Ideal theory in commutative semigroups - Springer Link

Semigroup Forum Vol. 30 (1984) 127-158 9 1984 Springer-Verlag New York Inc.

SURVEY A R T I C L E

IDEAL THEORY

IN COMMUTATIVE

SEMIGROUPS

D. D. Arlderson and E. W. Johnson Comm~icated

by Boris M. Schein

INTRODUCTION We use the t e r m semigroup semigroup

w i t h 0 and I.

empty subset

I ~ S

The m u l t i p l i c a t i v e developed search

are many

structural

The purpose between

of rings.

This

and

called

of this paper

groups

(called

Noetherian Noether

just the lattice In Section

and hence

2).

which

generally

Both

a local of an

includes

Noetherian properties

N-semigroups

of ideals

distributive

of semi-

properties

of semigroups

have ideal-theoretic rings.

semigroups

a class

which have ideal-theoretic This class

and dif-

are m u l t i p l i c a t i v e

We define

rings have lattices

of ideals

simpler

and the ideal theory

(with 0 and i adjoined).

In fact,

and that

ring t h e o r y

the

by the fact that both

structures

rings.

commutative

commutative

lattice.

However,

of semigroups

(see Section

N-semigroups)

close to Noetherian

from commutative

theory.

re-

There

of semigroups

is to study the similarities

rings have ideal

semigroups

i E S).

are possible.

the ideal t h e o r y

r-semigroups

since

ring is a h i g h l y

to by the numerous

allow m u c h more freedom

very close to commutative cancellation

SI = I

is a non-

[15],[29],[35],[37]).

and theorems

in semigroup

x-systems

S

in a commutative

the ideal theory

study is m o t i v a t e d

and commutative lattices

(and hence

as m a y be attested

results

multiplicative

in a semigroup

(for example,

between

for a semigroup

only weaker

ferences

SI ~ I

and books

analogues

I

of ideals

Many definitions

have natural

groups

with theory

similarities

of rings.

axioms

An ideal

area of research

articles

to mean a commutative

that

Noether

r-semivery

and are a

lattice

N-semigroup.

2 we define the necessary

terms~

give some basic

is

ANDERSON[JOHNSON

results ideals

about m u l t i p l i c a t i v e of a semigroup.

semigroup.

conditions.

r-semigroups

bibliography rings,

and

that

In Section

N-semigroups,

of the relevant

and m u l t i p l i c a t i v e

and look at the lattice

As m e n t i o n e d

about

i.

An ideal

If

I

and

I J

in

S

IJ = [iJ I i E I, j E J3 _ ~ [I _

is a nonempty

also ideals.

~(S),

set-theoretic

plete

completely

adjoined [03 of

S

I

of

to any semigroup.

S.

(u) = S. unique

collection

is a finite ... V A

u E S

S.

Further,

an element [A~3

L

NI

S,

if

are

is a comare the

~(S)

both

is a com-

that

ideal

O E S

gives that

[03

simply

Sl = I

a unit

as O.

We

for each ideal

if there

is a unit group

of ideals

of a lattice.

gives that

u

a 0 and a i is

a 0 and i can be

any intersection

element

is called

Hence

ideal of

subcollection

contains

Hence

form an abelian

L

and of

of sets,

The assumption

i E S

uv = I.

In a lattice nonempty

that

maximal

SI ~ I.

and

UI

of ideals

S

We often write the

The units

(proper)

then

0 and

More generally,

since as is well known

ideal of

An element

such that

with

I N J

S.

with

lattice.

that a semigroup

The assumption

of

and intersection

in generality

is nonempty.

I ~ S

I U J,

of ideals,

will also use 0 to denote the least

v E S

subset

then

the lattice

distributive

is the smallest

commutative

since the join and meet operations

union

Our assumption really no loss

semigroups,

multiplicatively

are also ideals

In fact,

usual

S,

collection

Hence

plete lattice.

of

We give a large

we will use the t e r m semigroup

written

is a nonempty

are ideals

few or

to the cases

2.

in the introduction, semigroup

with

of a

theory.

SECTION

to mean a commutative

can be obtained 4 we specialize

respectively.

papers

lattice

of

3 we look at the ideal theory

We focus on the results

no cancellation of

lattices

In Section

exists

a

if and only if

and the nonunits

form the

S. A

is called

of elements [A i ,

,An3

of

compact

if for any

L

with

A ~ VA

of

[A 3

with

is said to be compactly

,

there

A ~ A IV

~enerated

if each

n element and

of

L

a E S.

is a join of compact Then

clearly

(a) = Sa = [sa I s E $3 S

is compact

elements.

the p r i n c i p a l

is compact.

More generally,

if and only if it is finitely

128

Let

S

be a semigroup

ideal g e n e r a t e d

by

a,

an ideal

generated,

that

is,

I

of

ANDERSON/JOHNSON

I = (al,...,a n) = ( a l ) U ...U ( an) for some al,. " ' ' a n E S. Since each ideal of S is a union of principal ideals, ~(S) is compactly generated. ~(S)

Hence

~(S)

is a so-called algebraic lattice, that is~

is a complete compactly generated lattice. A multiplicative

lattice is a complete modular lattice

which there has been defined a commutative,

tion w h i c h distributes over arbitrary joins (i.e., and has greatest element

I

L

on

associative multiplica-

as a multiplicative

A(VB ) = VAB )

identity.

Both the

lattice of ideals of a semigroup and the lattice of ideals of a commutative ring form multiplicative

lattices.

(By a ring we shall al-

ways mean a commutative ring with identity.) For ideals

J

and

K

in a semigroup

ring) we define the residual of J: K = {s E S I sK ~ J~. L,

A: B = V[X ~ L I XB ~ A]. L

S

b_~ K

More generally,

one defines for elements

greatest element of

J

A,B E L,

Clearly

(or in a commutative

to be the ideal for a multiplieative

lattice

the residual

(A: B)B ~ A

and

A: B

is the

with this property.

In [58], M. Ward and R. P. Dilworth extended the Noether decomposition theory to suitably defined multiplicative

lattices; however,

further development was not possible because of the lack of a proper abstraction of principal ideals, principal element. multiplicative AACB And

lattice

= C((A: C ) A B ) C

is weak meet

( A v (0 : C) = CA: C) that a meet Finally,

C

In [23], Dilwortb defined such a

Following Dilworth we define an element L

to meet

( A V (B: C ) =

( A C V B ) : C)

A E L.

AAC

for all

(Setting

is said to be principal if Since

L

C

is modular,

A,B ~ L.

= C(A: C) B : I

[join) principal element is weak meet

and join principal.

in a

(join) principal if

(join) principal if for all

C

(B = 0)

shows

(join) principal.)

is both meet principal an element

C

is princi-

pal if and only if it is both weak meet principal and weak join principal [19]. We shall often use the following two facts. memt of a m u l t i p l i c a t i v e

lattice

L.

if and only if w h e n e v e r

B ~ C

for

D E L

with

B = DC

(we may take

principal if and only if whenever A m B V (0: C).

Then

C

be an

ele-

there exists an element

D = B: C). for

Also,

principal if and only if it is a multiplication

C

A,B E L

Thus in a commutative ring an ideal

129

C

is weak meet principal

B ~ L,

AC K BC

Let

C

ideal.

is weak join we have is weak meet (For the

ANDERSON/JOHNSON theory of m u l t i p l i c a t i o n

ideals,

see [2] and [5].)

The weak join

principal condition should be viewed as a cancellation property. fact, an ideal in a commutative ring is a cancellation

ideal if and

only if it is weak join principal and has zero annihilator. theory of cancellation ideals, see [6] or [29].)

In

(For the

For further facts

about principal elements the reader is referred to [3]. For a commutative ring

R,

a principal ideal

seen to be a principal element of to b e i n g true.

An ideal

if and only if

A

ideal of

HM

A

of

~(R). R

is easily

The converse is very close

is a principal

is finitely generated and

for each maximal ideal

(a)

M

AM

of

R

element of

is a principal ([47]).

Hence in an

integral domain, a nonzero ideal is a principal element of and only if it is an invertible ideal. (i.e.,

R

~(R)

is a quasi-local

if ring

if and only if it is principal.

For a semigroup

S,

the situation is more complicated.

of

S

is meet principal:

for any ideals

J

and

principal ideal = J(a) N K

R

~(R)

has a unique maximal ideal), then an ideal is a principal

element of

then

If

~(R)

y 6 K

(a)

and

y = ja

y = ja E ( J A (K: (a)))(a). multiplicative

lattice,

K

for some

of

S.

For let

j ~ J.

N o w any

( J N (K: (a)))(a)

Then

y E J(a) N K,

j E K: (a),

so

Since the other containment holds in any

(a)

is meet principal.

However,

a princi-

pal ideal need not be weak join principal as the following example shows.

y = y

Let 2

S = [0,l,x,y] and

be the semigroup with multiplication

xy = yx : 0.

Then

but not weak join principal, :

(x)

and

(y)

are meet principal,

since, for example,

(x) U(y) = (x,y) ~ S = (x) : (x) = (x)(x) : (x).

to note that the n o n p r i n c i p a l maximal ideal

2 x : x ,

(x) U (0 : (x)) It is interesting

(x,y)

of

S

is meet

principal. Since confusion might otherwise arise on occasion, w h i c h is assumed to be principal both as an ideal of element of the m u l t i p l i c a t i v e

lattice

~(S)

S

an ideal

A

and as an

will be said to be

strongly principal. The following proposition

characterizes

principal ideals w h i c h

ar~ strongly principal. PROPOSITION o~f

S. (i)

2.1.

Let

S

b e_e[ semigroup and

Then the followin~ statements (a)

i__ssstrongly principal,

130

(a)

~ principal

are ezuivalent:

ideal

ANDERSON/JOHNSON (2)

(a)

i_~s $oin principal,

(3)

(a)

is weak join principal,

(4)

(a)(b) = (a)(c) # 0

(5)

ab = ac ~ 0

Proof.

implies

It was previously

always true.

(3) ~

(a)(b) # O,

(b) = (c).

J

= (J: (a)) U K. ya E (a)K. ya = ak then

b = Xc

Now

and Let

K

that

y = ~k E K

ways holds,

so

y E J : (a),

X E S.

(5) ~--~ (2):

(JU (a)K) : (a)

ya ~ J U (a)K.

Thus

in the second case

then

y ~ J: (a);

if

ak # O,

Since the other containment

al-

is join principal.

A multiplicative every element

we must show that

ak = O,

is

so we have

(4) ----~(5) is obvious.

S

If

for some

(a)

(c) ~ (b),

y E (JU (a)K) : (a),

k ~ K.

(2) ~ ( 3 )

implies that

(a) = (c) U (0: (a)).

Similarly

of

X E S.

(i)~=~(2).

(a) = ((c)(a)):

In the first case

for some

for some

(a)(b) = (a)(c)

(b) ~ (c).

The implication

For any ideals

(b) = (c),

observed that

(4).

(b) U(O : (a)) = ((b)(a)): Since

implies

lattice

is said to be principally

is a join of principal

elements.

6enerated

Similarly

a multipli-

cative lattice may be defined to be (weak) meet principally or (weak) join principally

generated.

if

generated

In a similar vein we have the

following proposition. PROPOSITION ~(S)

2.2.

For a semigroup

S,

the followin~

conditions

on

are e~uival,ent: (i)

~(S)

i__ssprincipally ~enerated,

(2)

~(S)

i_ss ~oin principally

(3)

~(S)

is weak 00in principally

(4)

for any

a 6 S,

ab = ac # 0

implies

(b) = (c),

(5)

for any

a 6 S,

ab = ac # 0

implies

b = kc

unit (6) Proof.

for any

proposition

a E S,

k E S.

that is,

While a principal of

(2) =~ (3) follow from definitions.

(3) ~--~ (4) follows

lows from the previous

element

for some

i_ss strongly principal.

immediately

from the previous

since it is easily seen that a principal

for some

(k) = (i),

(a) (I) ~

pletely join irreducible. b = kc

generated,

~ E S,

The implications

The implication

generated,

~(S),

k

(4) =~ (5): Hence

We have

(b) = (c)

kac = ab = ac = lae # 0,

is a unit.

proposition.

The implication

element

131

so so

(5) =~ (6) fol-

(6) =~ (i) is obvious.

ideal of a semigroup

a principal

ideal is com-

of

S need not be a principal ~(S)

must be a principal

ANDERSON/JOHNSON

ideal. THEOREM 2.3. of

S.

~(S)

Hence the principal elements of

ideals

(a)

implies # 0

A principal element of

o_ff S

b = ~c

implies

Proof.

Let

~(S)

for some

k E S.

ideal

are just the principal

that satisfy the cancellation

law:

(Or equivalently,

ab = ae ~ 0 (a)(b) = (a)(c)

(h) = (c).) J

be a principal element of

(j) = (j) n J = J((j) : J). = J(G~((j)._~j : J)). J = (j).

must be a principal

If

Otherwise

Hence

J =

~(S).

For

j E J,

U (j) = U J((j) : J) jEJ j6J

J~J~((J) : J) = S, U ((j) : J) ~ M,

then

some

where

M

(j) : J = S,

so

is the maximal ideal

jEJ of

S.

In this case

S = M U ( 0 : J).

Hence

J = JM.

Since

S = (0: J),

J

is weak join principal,

so

J = 0.

Thus

J

is principal.

The last statement of the t h e o r e m follows from Proposition

2.1.

In [23], Dilworth defined a Noether lattice to be a multiplicatire lattice satisfying the ascending chain condition in which every element is a (finite) join of principal elements.

A Noether lattice

is an abstraction of the lattice of ideals of a Noetherian tive ring.

commuta-

In the same paper Dilworth extended the Noether decompo-

sition theory, the Krull Intersection Theorem and the Principal Ideal Theorem to Noether lattices.

For the development

of the theory of

Noether lattices, the reader is referred to [1],[7-12],[18-19],[23], and [32-33]. In [3], A n d e r s o n introduced

r-lattices

lattice of ideals of a commutative ring. plicative lattice

L

as an abstraction of the

An

r-lattice is a multi-

that is compactly generated,

principally

ated and has greatest

element

I

compact.

A multiplicative

lattice

L

is said to be quasi-local

has a unique

(proper) m a x i m a l element

M ~ I

This is easily seen to be equivalent to X K M.

Hence for any semigroup

S,

pletely distributive m u l t i p l i c a t i v e

and

X ~ L

~(S)

I

with

gener-

if

L

is compact. X ~ I

is a quasi-local

implies com-

lattice in w h i c h every element is

a join of completely join irreducible weak meet principal (namely, the principal ideals) that are compact.

elements

What is somewhat

surprising is that the converse is also true. T H E O R E M 2.4 ([3, Lemma 3.1]). then

L

Let

L

be ~ m u l t i p l i c a t i v e

lattice,

is isomorphic to the lattice of ideals of a semigroup if and

only i f

132

ANDERSON/JONHSON

(A)

L

is distributive,

(B)

L

i_~s Quasi-~ocal,

(C)

there L

exists

which

ucts,

r-lattice. S

a set

S

of weak meet principal

L

under joins,

S

Thus

an

S

implies

T

T

0

sults from commutative

a distributive

~(S)

2.5

([3, T h e o r e m

is a ~uasi-local

the lattice

of ideals

dition.

an

is an

if and only if

of a cancellation (necessarily

and a

i

is an

~ E S.

adjoined

without if

r-semigroup.

satisfying

T

An

O)

doesn't r-semi-

a weakened

condition

semi-

cancella-

necessary

for re-

is an

r-semigroup~

r-lattice.

It follows

then

~(S)

is

from the previous

is also true. 3.2]).

Let

S

distributive

be an

r-semigroup,

r-lattice.

r-lattice.

of an

then

Conversely, L

Then

let

L

i_~s isomorphic

to

r-semigroup. r-lattice

the ascending

N-semigroup

Theorem

S

seen that an

only if it satisfies r-semigroup

if

distributive

It is easily

~(S)

ring t h e o r y to carry over to semigroups.

quasi-local

be a Quasi-local

for some unit

cancellation

that

t h e o r e m that the converse THEOREM

S

and compact.

if

r-semigroup

semigroup

Then

of

law:

b = Xc

as a semigroup

exact

We have observed

is an

adjoined

already have an identity.

tion condition--the

r-semigroup

is a generalization

with

group should be v i e w e d

an

S

be a cancellation

be

elements

is closed under prod-

are join irreducible

cancellation

r-semigroup

Let

and let

2.2,

the following

ab = ac ~ 0

group.

elements

will be called

By Proposition

satisfies

and

generate s

and whose

A semigroup

7

chain

is a Noether condition.

if it satisfies

lattice

We shall call an

the ascending

2.5 applied to the Noetherian

if and

chain

case yields

con-

the fol-

lowing theorem. THEOREM ~(S)

2.6

([i, T h e o r e m

is a distributive

tributive ideals

local Noether

of an

Theorem for

Let

S

local Noether

be an lattice.

lattice i_~s isomorphic

2.5 and T h e o r e m and Noether

Then

Conversely,

any dis-

to the lattice

of

2.6 allow us to apply the known results lattices

to

This will be done in Section

3 we see what

N-semigroup.

N-semigroup.

r-lattices

groups.

2]).

can be done in more

will be given to p r i m a r y

4.

r-semigroups First,

generality.

decompositioms

133

and

however,

Particular

N-semiin Section attention

and related properties.

ANDERSON/JOHNSON

We mention in passing that the ideal theory of a commutative ring has also been abstracted in a different manner--the theory of x-ideals as given by K. E. Aubert [17]. system. that

Let

S

S

has a

We briefly define an

x-

be a commutative semigroup (here we do not assume 0

or

i).

We say that there is defined an

S

if to every subset

of

S

there corresponds a subset

of

S

such that for any subsets

A

and

A ~ Bx =~ Ax ~ Bx,

and

A

x-system

in

B

of

ABx ~ Bx N (AB)x.

S

we have

If we take

x A ~ Ax,

Ax = SA U A

we have the usual notion of the (semigroup) ideal generated by (which Aubert calls an Aubert defines

s-ideal).

A

A

The previously mentioned paper by

x-systems, gives many diverse examples, and develops

the elementary theory of tinued the study of

x-systems.

Aubert and others have con-

x-systems in a series of papers.

reader is encouraged to consult these papers. in [16], Aubert had observed that the

The interested

We remark that already

s-ideals of a semigroup form a

complete distributive residuated lattice and that the parts of the theory of ideals in rings was application to semigroups ideals. Aubert's

x-systems generalize some earlier work on abstract ideal

systems by H. Prefer [49] and P. Lorenzen [40-42]. SECTION 3. We maintain our convention that written multiplicatively with

0

S

and

is a commutative semigroup i.

In this section we inves-

tigate which results from commutative ring theory carry over to semigroups without the further cancellation assumption that the principal ideals

(a)

of

S

are strongly principal.

The calculus of ideals of a semigroup is very much like that of a commutative ring. easily verified.

The following ideal-theoretic relations are

Here

A, Ai, B, B i

and

AB=BA A(BC) = ( ~ ) C SA=A 0A=

0

A(uis i) = UAS. 9 i z AB ~ A N B

A(BA C) ~ ABNAC (A : B)B ~ A A c (A: B)

134

C

denote ideals of

S.

ANDERSON/JOHNSON

(nAi)i : B = N(A. : B] 9 i I (A : B) : C = A : (BC) A: (VB i) = 0(A:~ B.~) A:B A proper ideal a E P if

or

(p)

h E P.

P

of

S

is called prime if

An element

is a prime ideal.

= A: (AUB).

p ~ S

ab E P

implies

will be called a prime element

Perhaps the greatest divergence between

the ideal theory of semigroups and that of rings is due to the next simple proposition which stands in contrast to the following wellknown and useful result from commutative ring theory: ideal of a commutative ring

R

Ii,-..,l

with at least

I ~ I.

are ideals of

n for some

R

and

I ~ flU .--UIn n-2

If

I

is an

(n~2)

where

of them prime

then

i.

i

PROPOSITION 3.1.

Let

S

be a semigroup and

lection of prime ideals of Proof. P. l0

Let

ab E ~Pi"

is prime, say

S.

Then

[Pi~

a nonempty col-

Then

UP. is a prime ideal of S. i i ab E Pio for some i 0. Hence since

a E P. ~ UP.. m0 m

Thus

UP. l

is prime.

While an arbitrary intersection of prime ideals need not be prime, an intersection of a totally ordered family of prime ideals is still prime. prime ideal P0 Q

An easy application of Zorn's Lemma gives that any P

containing

minimal over with

A,

A

i.e.,

can be shrunk to a prime P ~ P0 ~ A

PO ~ P

with

and there is no prime ideal

P0 ~ Q ~ A.

Given an ideal

A

= Is E S I sn ~ A ideal containing

A.

to be semi~rime) if

of

S

we define the radical of

for some natural number An ideal ~

= A.

A

n].

A

Clearly

to be ~fA

is an

is said to he a radical ideal (or

Many familiar properties of the radical

for rings are also true for semigroups.

If

A, B, B.

are ideals of

i

S,

then ~

=~,

~-~

: ~/A n ~

=~/~B,

and ~J-~i = ~B~-i .

From

the last equation it follows that the union of radical ideals is a radical ideal. In a commutative ring with identity, and in many other algebraic systems, the radical of an ideal is the intersection of the minimal primes containing it. subset S

if

T

of

0 ~ T

S and

This is also true for semigroups.

A nonempty

will be called a multiplicatively closed subset of a,b E T

implies

135

ab ~ T.

ANDERSON/JOHNSON LEMMA 3.2 (Krull's Lemma for Semigroups). let

T

be a multiplicatively

ideal of

S

with

ANT

= r

Let

S

be a semigroup,

closed subset of

S

and let

A

be an

Then there exists a unique ideal

maximal with respect to the property that

B N T = r

B ~ A

Moreover,

B

i_~s prime. Proof.

We note that this version of Krull's Lemma is stronger than

the ring version,

since in the latter case, the ideal

general, unique.

Nevertheless,

establish:

simply let

B

that are contained in that

B

ments

is prime. x,y E S

have

Let

B

is not, in

is trivial to

be the union of all ideals containing

B

is not prime, then there would exist elex ~ B

with

and

y ~ B.

(BU (x)) N T @ r

be a semigroup and

A

But then we would and

(B : x) N T ~ r

an ideal of

is the intersection of the prime ideals minimal over

S.

Then

A.

Proof.

Using Lemma 3.2, the theorem follows as for commutative

rings.

If

3.2,

plies

B

n

x ~ ~IAA, then with

~

T = Ix ]n=l

is a prime containing

A

and with

B

but not containing

as in Lemma

x.

A proper ideal Q of S is said to be primary if ab E Q ima E Q or b n E Q for some natural number n. If Q is pri-

mary, then

~

P-primary.

Thus an ideal

and

A

A standard technique from rings shows

xy ~ B,

S

B

S-T.

(BU(x))(B : x) ~ B

THEOREM 3.3. ,/A

If

with

the existence of

ab E Q

is a prime ideal.

implies

Q

a ~ Q

If

is

P = ~,

we say that

Q

P-primary if and only if

or

b E P.

Proposition

is

,/Q = P

3.1 may be gen-

eralized as follows. PROPOSITION

3.4.

sarily distinct)

Let

{Pi ]

be a family o f prime ideals

and for each

P.

let

Q.

i'

~

Pi

is a prime ideal and of

union

Proof.

UQ i

P-primary ideals is By Proposition

:

: Fi

3.1,

1

be

(not neces-

Pi-primary.

Then

--

is

UP.-primary.

--

i

In particular,

any

1

P-primary. ~Pi

is a prime ideal.

Suppose that

ab

UQ i

Then

Also

ab

Qi

for 0

some

i O.

Hence either

a ~ Qi

~ UQ i

or

0 ~Qi

is

b E Pi0 ~ UP.. ~

Hence

UP.-primary. 1

It is easily seen that if Q I A "-" N Q n family of

is

P-primary.

QI""'Qn

Of course if

are [Qi }

P-primary ideals,

NQi

need not be

it is not hard to prove that

NQi

is

i~--~i = P.

136

P-primary, then is an (infinite) P-primary.

In fact,

P-primary if and only if

ANDERSON/JOHNSON A semigroup

S

will be called Noetherian if it satisfies the

ascending chain condition on ideals.

It is easily seen that

S

is

Noetherian if and only if any nonempty collection of ideals of ~as a maximal element or if and only if every ideal of ly generated.

S

S

is finite-

As with rings, it is enough to check to see if the

prime ideals are finitely generated.

As we will see later, under

somewhat more restrictive conditions, it is even sufficient that the maximal ideal be finitely generated. THEOREM 3.5 (Cohen's Theorem for Semigroups).

A semigroup

S

i~s

Noetherian if and only i f every prime ideal is finitely generated. Proof.

One way is clear; so assume that every prime ideal of

finitely generated.

Suppose that

Lemma, there exists an ideal

P

Suppose that

ab E P,

and

a ~ P

(Xl,''-,x n)

P

so

Xl~''',X s

P

J

b ~ P.

Then

and

(Xs+l,..-,x n)

Then

Thus

But then

or

r E J,

P = (Xl, 9 .-,x s,Ja)

This contradiction shows that

so

J = P : (a).

(P,a) = Then

Suppose that

Then

P = (Xl,..-,Xs,Ja).

y E (Xs+l,''',Xn)~

we are done, so suppose r.

is

y E (P,a) = (Xl,.-.,x s) U

y E ( X l , " ' , x s)

for some

generated.

Let

Xs+l,...,x n E (a)-P. y E P.

so

y E (Xl,''',x s)

(P,a) ~ P,

is also finitely generated.

For suppose that

S

By Zorn's

is not prime, so we have

is finitely generated.

(b,P) ~ J,

y = ra

is not Noetherian.

maximal with respect to not being

finitely generated. but

S

S

(a).

y E (Xs+l,'-',Xn), so

y E Ja

If

then

which is finitely

which is finitely generated.

is Noetherian.

Before proceeding with some ideal-theoretic results, we introduce some semigroup constructions.

First, we look at the localiza-

tion process for semigroups. Let of

S

S

be a semigroup and

T

a multiplicatively closed subset

(recall that this means that

ab E T).

We define

ST,

tiplication units in

for some

s/t; u E T.

s/t- se/t e = s J / t t ~

ST .

s E S, t E T; ST

T.

a_t_t T, where

and the elements of

to be the s/t N se/t ~

T

become

ST

and the primes of

S

There is also the usual correspondence between primary

S

of

T* = Is E S I ~ sl E S ~ s J E T~.

by

S

implies

is a semigroup under the mul-

ideals of T

a,b E T

We state without proof that there is a one-to-one

correspondence between the primes of avoiding

and

the localization of

set of equivalence classes ~=~ ust e = ust e

0 ~ T

and

ST .

As for rings, we define the saturation

137

T*

Then using Lemma 3.2 as

ANDERSON/JOHNSON for rings we get that

S-T*

is a union of prime ideals.

However,

since in a semigroup a union of prime ideals is prime, we get that every saturated multiplicatively for some prime by

P.

closed set has the form

As for rings, we have

S T = ST.

T* = S-P

which we denote

Sp. By a homomorphism,

operations and

0

and

we will mean a map preserving the semigroup I.

It is clear that there is a one-to-one

correspondence between surjective homomorphisms define a special congruence. ideal.

Let

S

We define the semigroup

with multiplication group and the map

semigroup, or an

if

~f

S

Clearly

given by

~(x) = r ~

is a Noetherian

N-semigroup,

is lattice isomorphic to

xy ~ I

x~EI"

> S/I

semigroup homomorphism.

be a semigroup and

S/I = Ix E S I x E S-I

xy = In xy ~: S

and congruences.

then so is

~(S)/I

I ~ S or

S/I

is a semi-

x ~ E I an

A

an ideal in

if we can write

mary ideals, say

S. A

write

A = Q I N ... A Q n

after removing the

S

and those in

We say that

A

S/I.

Let

S

be

has a primary

where

Qi

is

~i

P-primary ideals is

where the

Qi's

~(S/I)

has a finite intersection of pri-

A = Q I N -.. A Q n

Since the intersection of two

r-

We note that

We next examine primary decomposition in semigroups.

decomposition

is a

and that we have the usual corre-

spondence between prime and primary ideals in

a semigroup and

an

x = 03

semigroup,

S/I.

We

with

= Pi-primary. P-primary, we can

P.'sl are distinct. ~iQj ~ Qi'

Moreover,

we can write

J A = QI N .-. N Qn be removed. A.

where the primes Pi's

Such a decomposition

Without any further assumptions,

primes

PI,...,Pn

THEOREM 3.6.

Let

S

may of

it is easy to show that the

b__%e~ sen/group and A = QI n .-- N Qn

[PI'''''Pn ] = [P ~ P r i m e

x E S].

Qi

do not depend on the decomposition.

normal decomposition Then

are distinct and no

is called a normal decomposition

A

an ideal of

where

ideal of

Qi

is

S I P = ~:

Hence any two normal decompositions

havin ~

pi-Primary. x

o_~f A

S

for some

have the same

lensth and the same associated primes. Proof.

Suppose that

P = ~A i x

for some

x 6 S.

Then

P = ~/A: x =

d(aln .nanl:x:J(Ql:xln..-n(an:X)=J41:xa.. ndas ~I N ... N ~ n

where either

x E Qi ). Hence Conversely,

~.i = P'I

(if

x ~ Qi )

P = P. = P. for some i. 1 1 we show that, say, P1 = A~7~." x

138

or

~.i = S

for some

(if

x ~ S.

ANDERSON/JOHNSON Since

~

A = QI ~ "'" N Q n

Q 2 N ... nQn.

Let

N (Qn:X) = QI: x.

is a normal decomposition,

x E Q 2 N ... NQn-A. Hence

~

Then

A = Q I N ... A

Qn

A: x = (Ql:X)N ...

=~#QI: x = PI

since

QI: x

is

PI-

[PI,.-.,Pn]

the

primary. As in the ring case, we call the prime ideals associated primes of

A

and write

Ass(S/A) = [PI,-.-,Pn].

While

the primes associated with a normal decomposition are unique, the primary ideals themselves need not be. free semigroup on integers].

Then

composition for

X

and

Y:

For example, let

S = [0 5 U [xnyml n,m

(X2,Xy) = (X2,Xy,Y n) A (X) (X2,Xy)

for each

n ~ I.

the following uniqueness result.

Let

S

ideal of

S.

The

ideal

S

and

A

an ideal of

A(P) = Ap N S = [x E S1 rx E A

THEOREM 3.7. ideal

A

Let

A = Q I N ... n Qn

in the semigroup Then

S

composition.

As for rings, we do have be a semigroup,

P

a prime

P-component of

A

is the

where

A

r E S-P~.

be a normal decomposition for the Q.

is

P.-primary.

- -

and

P

Let

P m A

l

N[Qi I P i ~ P] = A(P)

N[Qil Pi ~ P] depends only on

be the

is a strong normal de-

for some

I

be a prime ideal.

S

nonnegative

and hence the ideal

and not the particular de-

If

P. is a minimal prime ideal of A, then any two z normal decompositions of A have A(P i) as the P.-primary c o m ponent. In the previous example, we saw that number of normal decompositions. (X,Y) unique

and

(X).

(X)

(X2,Xy),

(X)

is the

It is not hard to show that if

is any normal decomposition of

(X,Y)-primary, then

Q ~ (X2,y).

Hence

P-primary ideals is

(X2,Xy)

where

Q

(X2,Xy) = (X2,y) N (X)

is the unique "maximal" normal decomposition of that a union of

had an infinite

Here the associated primes are

is minimal over

(X)-primary component.

(X2,XY) = Q N (X) is

Since

(X2,Xy)

(X2,Xy).

The facts

P-primary and that the lattice

of ideals of a semigroup is completely distributed allow us to define a canonical normal dec ompositipn for an ideal in a semigroup which has a primary decomposition. THEOREM 3.8.

Let

A

be an idea_____~lin ~ semigroup

decomposition with associated primes Pi-primar~ ideals A

=

QIN

-.. N Qn

Qi*

with

A = QI* N

[PI,...,Pn]. 9 ""

N Qn*

havin~ a normal Then there exist

such that if

i~s any normal decomposition o~f A

where

Qi

is

Pi-primary, then Proof. "

Qi ~ Q*" i For each prime P.

S

i

let

Q* = U[Q is i

139

P.-primary I Q m

occurs

ANDERSON/JOHNSON in a normal decomposition

of

A~

Then

Q*

9

primary ideals is

P.-primary.

A ~ QI* N . . . N Q n ~ .

Clearly *

x E Qi'

For each

i,

mary and

A = Qli ~]Q2iN . - -.A Q.i i.A

so we have

A.

Then

Let

x E Qii

x E Q t3...NQn.

where

Qii

is

P.-pri-?_

NQni

is a normal decomposition

x E (j:l [j Q'• ) ~ "'" f] (j~iQnJ)"

Thus by induction we are

n

of

P.1

We are done once we show that

1

* A = QI* f l . "" N Qn"

being a union of

i

n

done if we can show that if

A = QI N ...~]Qn = QIe n -.-fiQan

are

normal decompositions

where

Qi,Q~

then

If

PI '''''Pn

A = (QIUQI)f]-. 9

of

A

(QnUQen).

are

Pi-primary,

are all minimal over

# . " 'Qn = Qen QI = QI'"

then by the previous theorem

A,

and we are done.

Suppose that some

P. is not minimal over A. After renumbering the J P.'sm if necessary, we can assume that PI,..-,Pj_I are the primes minimal over

A

and

P.j

is a minimal element of the set

[Pj ,... ,Pn].

Then by the previous theorem, ~e have

Q I N .-- O Q j- I N Q ~ j = QI N " - . N Q j _ I N A ( P j) = QI n - . - N Q j _ I A Q j so by the distributive law we have

QI f] . - . O Q j _ I D Q j

= QI A ... O Qj_I N Q0j. Pj+I'''''Pn'

If

if necessary,

[Pj+I'''''Pn ]"

= Q1 n ...nQj_if] ( Q j U Q ~j)

j = n,

we are done, otherwise renumber

so that

Pj+I

As before we get that

is a minimal element of

Q I N -.. N Q j _ I N Q j N Q j + I ~ f]Qj_l n Qj' N QJ+I" ...n (QnUQtn).

= QI r] ..-~]Qjf] (QjUQ~.) f] (Qj+IUQj+I) = Q1 N 9 Continuing we get A = QI N . . . O Q n = ( Q I U Q I ) N Let

S

be a semigroup and

A

an ideal of

decomposition with associated primes partial order on the set

•

S

PI'''''Pn"

having a normal We can define a

of normal decompositions

A

by

QIA ... A Q n < QlS N --. nQen

(where

if

The proof of Theorem 3.8 shows that

Qi ~ Q~" for each

a lattice with meet

i.

Qi'Q~ are

of

(Q113 " " N Q n )

P.1-primary) if and only J

is

A (QI N ...nQan) = (QINQI) O .-.

N (QnnQn) and Join (Q113 " " N Q n ) V (QI N . . . D Q n) = ( Q I U Q I ) N ... A (QnUQtn). In fact, J is closed under arbitrary joins and J has as greatest element

QI Q " " Q Q * n

occurs in a normal decomposition We call

c E S

with

cd E A.

we have

Z(S/A) = PI U . . . U P n

a prime ideal9 divisors,

~i

Q*i = U[Q A

if there exists a

We denote this set by where

is Pi-primaryl Q

A~.

a zero divisor mod

d E S-A

decomposition with

where of

Z(S/A).

A = QI N .-. N Q n

= P'I" Here, however,

As for rings is a normal

P1 U ..-U Pn

itself is

Thus, even though a prime ideal consists of zero

it need not be contained in an associated prime.

140

ANDERSON/JOHNSON For an ideal

A,

if

exists a natural number is Noetherian,

~/A n

is finitely generated,

with

(~)n

~ A.

then there

In particular,

is

Q

is said to be strongly

P-primary,

and if

pn ~ Q

for some natural number

P-primary.

n,

is said to be a stron~ normal decomposition

Qi

Pi-primary.

is strongly

Ass(S/A)

= [P

If

A

then

if each

has a strong normal decomposi-

a prime ideal of

S IP = A :x

for some

(In other words, we can dispose of the radical.)

We next record the fact that in a Noetherian ideal has a strong normal decomposition. is said to be irreducible S)

If

A normal decomposition

A = Q l n ... A Q n

x E $I.

S

then each ideal contains a power of its radical.

Q

tion, then

if

implies

I = J

THEOREM 3.9.

Let

or S

if

I = J N K

semigroup,

every

Recall that an ideal (for ideals

J

and

I r S K

of

I = K. be a Noetherian

ible ideal is primary.

semigroup.

Then any irreduc-

Thus every ideal has a stron~ normal decompo-

sition. Proof:

A general p r o o f for multiplicative

and Dilworth

[58].

lattices

is given by W a r d

A simplified proof for semigroups

appears in [16].

However, the proof is sufficiently short that we include it. is irreducible

and

ab E Q

Q: b ~ Q : b 2 ~ Q: b 3 ~ "'', Q: b n = Q : b n+l .

Thus if

then

Q ~ Q : b. n

Since

~with

Q = ( Q U (b n)) N (Q : bn).

S

is a Noetherian

semigroup and

has a strong normal decomposition

P.-primary.1 =

a ~ Q,

we can choose an

Q

Hence

Q = Q U (bn),

b n E Q.

so

A

Then

with

If

[P

The

PI,.--,Pn

a prime ideal of

= P I U "'" U P n . divisors

is an ideal of

A = QI N 9 .. N Qn

are nnique since

S IP = A :x

A

for some

where

Qi

S, is

[PI,...,Pn] : Ass(S/A) x E S]

and

Z(S/A)

In a Noetherian ring, any ideal consisting of zero

is actually annihilated by a single element.

tainly not the case for semigroups,

since

P I U -'' U Pn

This is ceris a prime

ideal that will be annihilated by a single element if and only if it is an associated prime. Seeing that an ideal having a normal decomposition may have more than one normal decomposition, ize the semigroups

it is natural to attempt to character-

in w h i c h every ideal has a unique normal decompo-

sition.

A slight m o d i f i c a t i o n

that an

r-semigroup

S

of the methods used in [33] yields

has the property that every ideal has a

unique normal decomposition

if and only if every ideal is primary if

141

ANDERSON/JOHNSON and only if either

S

has only one prime ideal or

the only prime ideals of

0

and

M

are

and

K

S.

A standard argument using primary decomposition gives THEOREM 3.10 (Krull Intersection Theorem). a Noetherian semigroup Proof.

S,

See, for example,

For ideals

J

we have

~ jnK = J ~ jnK. n=l n=l [48, page 49].

Actually Theorem 3.10 follows from the Artin-Rees semigroups which we give next. indeterminate, tiplication

we define

If

S

S[x] = [sxml s E S

(sxi)(tx J)- = stx i+j

implies

S[x]

is Noetherian.

Xl,.-.,x n

where

and

S[Xl,.-.,x n]

S

is an

with mul-

A proof similar, S

Noetherian

is a Noetherian semigroup

any homomorphic

image of

is Noetherian.

THEOREM 3.11 (Artin-Rees Lemma for Semigroups). Noetherian semigroup and let exists an inteser Proof.

x

i a 0]

s,t E S.

Hence if

are indeterminates,

Lemma for

is a semigroup and

but simpler, to the Hilbert Basis Theorem shows that

and

in

N

A

so that

and

B

Let

S

be ideals of

A N B i = (AN BN)B i-N

be a S.

Then there

for all

i ~ N.

While the notational translation is tedious and several type-

setting errors make the proof difficult to read (in several instances "4"

or

ii].

"~"

should be

T! ~ TT) ,

~

this is a corollary of [34, Theorem

An alternative proof quite similar to that given by Rees

is to show that if

B = (bl,-..,b n)

then the semigroup

S[blt,...,bnt,t-l]

and

t

[51],

is an indeterminate,

is Noetherian.

However, this

easily follows from the remarks made in the previous paragraph. Without the assumption that

S

is an

r-semigroup,

esting but weaker results are still possible. the ideal theory of composition. ideal of

S

We are interested in

if every ideal has a unique strong normal de-

However, for the moment, we only assume that every

S

has a primary decomposition.

For the remainder of this section we assume that group with maximal ideal

M.

For an ideal

~A (S) : ~A = [QI Q is and We set

some inter-

A

M-primary and

~A (s) : ~A : ~x ~ S l Q ~ U A : (x)UA #(S) = ~0(S) = ~

from context.

and

We note that

of

~(S) = ~0(S) = ~ ~A = [x E S I M x U A

THEOREM 3.12 (Weak Intersection Theorem).

142

S,

S

is a semi-

we define

A =

for every Q ~ ~A ). when

S

is clear

= (x)UA~.

Assume that

A

is an

ANDERSON/JOHNSON ideal of the semigroup has a primary

Proof.

Clearly

mary.

Consider

n Qn"

Since

If

Qi

Thus

is

a primary

Q

is

an arbitrary

implies

then since

ideal,

x E A U ~.

Let

and

S

(BUQ)

and hence

~et

A

q

Qi

M-pri-

is

M-primary.

x E n~ A ~ Qi"

(x) = Qx.

Since

r-semigroup,

x = 0.

he

QxUA = Q I N ...

or

we have

or

ring or an

be a semigroup

of

S,

primary.

A = B U CM

A = B U CM

A = B U Q

containing

Hence,

Q

then

was

Mx = (x)

we focus on the

S/~0.

i_~s strongly C

A ~ Qi'

M-primary

of

Proof.

x E Qi

x E A

ideal theory

S

QxUA,

of

either

either

LEMMA 3.13.

A,B

x ~ n~A.

Hence

is a quasi-local

~

~et

decomposition

M U (0 : x) = S,

ment of

every ideal of

Then

M-primary,

M-primary,

S

and that

AU~ =~A ~n~A"

x ~ Qx CA.

If

S

decomposition.

implies

for all

in w h i c h

Then

~ M n = 0. n=l

implies

A = B.

A = B U CM n

Q E 5,

since

= B U (N~) = B U ~

d7 = 0

Q

Hence

for all

n,

is strongly

and since

~ = 0,

and every

ele-

for ideals

and hence

primary.

Since

the result

follows.

QE~ LEMMA n=l

3.14.

M n = 0.

S

maximal

M = (x).

is trivial

and

ideal,

B B

ideal of

S

of

S

and

decomposition

every normal

ideals

a sublattice

of

S

of

which ~

to

decomposition

of

A

ideal

S,

~(A)

of

in

is strongly

and

principal

and

ideal of

M m. in

Then

Mx n,

A

involving

an of

an

S

by setting M-primary

is a sublattice

143

(x n) = B.

It

j = k.

of

which has a ideal.

involves

3.8, the set

joins.

B = B N Mn

so

M-primary A

in some normal

~(S)

and choose

implies

an ideal of

is closed under

involves

S

principal.

decomposition

involved

of ~(S)

tend the domain

A

i s principal

(xi)(x j) = (xi)(x k) r 0 are strongly

and by the proof of T h e o r e m

primary

S

M

be any nonzero

is c o n t a i n e d

be a semigroup

(primary) 3.6,

B

is not contained

to see that

Hence the ideals

Theorem

Let

such that

= (B: (xn))x n

normal

in w h i c h

o_~f M.

Let

Let

be a semigroup

Then every nonzero

is a power Proof. n

Let

~(A)

an

of all

decomposition

arbitrary ~(A) ideal. S

of

joins.

= [S]

By

M-primary MA

is

We ex-

if no normal

Hence,

closed under

for any arbitrary

ANDERSON/JOHNSON THEOREM 3.15.

Let

S

be a semigroup in which

ideal has a primary decomposition. ~A)

@ = 0

and every

If, for every ideal

A

o_ff S,

has minimal elements, then every nonzero prime ideal of

maximal. o_~f S

If

0

is prime in

S,

S

i_ss

then either every principal ideal

i_~sstrongly principal or no

proper

(nonzero) ideal of

S

is

strongly principal. Proof.

Let

Q~ r (d). tains Let

d

be any nonzero element of

Choose

Every term of a primary decomposition

d

or lies in

Q0

S.

3.

Hence,

Qd

be the minimum element of

has an ~(Qd).

of

Q E ~ Qd

either con-

M-primary component. Then (Theorem 3.12)

= n~Qd = Qd. Hence Qd E 3, and therefore (d) E 3. Q0 that M is the only nonzero prime of S. Now, set P = Ix E S I xa = xb of

S,

for some

a ~ b].

so

Clearly

P

It follows

is a prime ideal

so the result follows.

We note that the hypothesis

of Theorem 3.15 is trivially satis-

fied if every ideal has a unique normal decomposition

and

@ = 0.

Gilmer [27] has shown that a quasi-local ring in which every ideal has a unique normal decomposition 3.15.

satisfies the conclusion of Theorem

Our proof is simpler and works equally well for quasi-local

rings. COROLLARY 3.16.

Let

S

be a semigroup in which

J = 0

and every

ideal has a unique irredundant primary decomposition.

Then

a chain.

__isstron61y

If

M ~ M 2,

then every nonzero ideal of

S

~(S)

i_~s

principal and a power o_~f M. Proof.

If

Q2 ~ QI'

QI then

and

Q2

are elements of

Q = QI n Q2

is

~

possibly not in

3,

If (x) ~ (y) or

x E M-M 2

or

(y) ~ (x),

form a chain. ~(S) and

y

(y) ~ (x). so

with

QI ~ Q2

and

M-primary with the two distinct

irredundant primary decompositions the elements of

~

Q = Q Since

and 0

Q = QI N Q2"

Hence

is the only ideal of

S

is a chain. is a nonzero element of Since

(y) ~ (x).

x ~ M 2, Hence

M,

then either

it follows that

M = (x).

(x) = (y)

Then necessarily

~ M n = 0 (Lemma 3.13), so every nonzero ideal is strongly principal n=l and is a power of M (Lemma 3.1~). It was mentioned earlier that a semigroup is Noetherian if every prime ideal is finitely generated and that under suitable conditions it sufficed to assume that the maximal ideal is finitely generated. We demonstrate this now.

144

ANDERSON/JOHNSON THEOREM 3.17.

Let

S

be a semigroup in w h i c h

ideal has a primary decomposition, every ideal of

l__ff M

is finitely ~enerated.

every ideal of

S

i_~sprincipal.

Proof.

be a finite set of generators

G

a prime ideal. n.

Since

M

Then

P = (PNG)

U MID,

is finitely generated,

and hence that

P = (PNG),

since

and every

is finitely~enerated,

S

Let

J = 0

If

so

M

for

M

and let

P = (PNG)

UMnp

it follows that ~ = 0.

then

i__ssprincipal, then

Hence

P

be

for all

P = (PNG)

P

U

is finitely

generated. The ring-theoretic

version of Theorem 3.17 is given in [13].

In

a similar vein we have the following theorem. THEOREM 3.18.

Let

S

be a semigroup with maximal ideal

finitely generated and has Proof.

U Mnp

t i m n = 0. n=l

Then

S

all

n.

Thus

P =

~ ((PNG)UMnp)

P = (PNG)

= ( P N G ) U ( ~ Mnp)

n=l

= (P n G)

since

w h i c h is

is Noetherian.

As in the proof of the previous theorem, we have

for

M

n=l

= O.

~ ~ n=l

To get deeper structural results, cancellation properties group are necessary.

such as

S

similar to Noetherian rings,

being an

r-semigroup or

We next note that for a Noetherian

the semigroup versions of Nakayama's the maximal ideal of

S),

Lemma,

fi M n = 0 n=l

N-semi-

semigroup (where

M

is

and the Principal Ideal Theorem need not

hold. Let group

S = [0,a,l]

with

w i t h maximal ideal

Nakayama's Lemma and Let

S

ab = a 2.

a

2

= a.

Then

M = [0,a].

n=~M n = 0

S

Then

fail for

is a Noetherian M=M 2=M 3 .... ,

S.

be the semigroup with generators

Then the prime ideals of

the maximal ideal of

S.

generated,

S

ideals,

has rank 2.

S

are

0,

a,b

Since

and the relation

P = Sa

Since each prime ideal of

is Noetherian.

semiso both

0 ~ P ~ M

S

and

M = (a,b)

is finitely

is a chain of prime

I

ideal

M Sb.

However,

M

is minimal over the principal

Thus the Principal Ideal Theorem does not generalize to

arbitrary Noetherian THEOREM 3.19.

Let

semigroups. S

be a semigroup in w h i c h

0

is prime and every

nonzero prime ideal contains a strongly principal nonzero prime ideal. Then every nonzero principal ideal of

S

is a unique product o f

strongly principal prime ideals and hence is also strongly principal.

145

ANDERSON/JOHNSON

Proof.

Let

prime

ideals~.

In fact, is,

X = Ix r 0 I (x)

(x,y E M)

element

Then by Lepta P n X = ~. tains

X

y E M-X.

3.2,

This

(y)

principal

3.20.

nonzero S

is u n i q u e l y Let

Then every proper uct of strongly

S

since

X

subset

is saturated, that there

is saturated, to a prime

every nonzero

prime

principal

principal

~rime

of prime

ideal of

ideals,

S.

that is a

(y) N X = r

ideal prime

P

with

ideal

con-

if every nonzero ideals.

be a s t r o n ~ - ~ - s e m i g r o u p in which

nonzero

of

ideal.

a stron~-~-semigroup a product

principal

closed

Suppose

X

can be enlarged

principal

ideal

x,y E X.

Then since

is impossible

a strongly

of strongly

to prove that

implies

We call a semigroup

THEOREM

is a product

is a m u l t i p l i c a t i v e l y

it is s t r a i g h t f o r w a r d

xy E X

nonzero

Clearly

S

0

is prime.

is uniquely

and hence

strongly

~ prod-

principal.

n

Proof.

Let

ization

of

each

x r 0 (x)

be any nonunit.

into a product

Let

(x) =

of prime

~ P. j=l 0

ideals.

be the factor-

We first

P.

is a cancellation ideal. If C and D n n CP i = DP i, then C ~ P. = D ~ P., so that

show that

are ideals

of

S

I

with

j=l J zero

c ~ C,

cx = dx

(c) = H P ~

and

into prime

ideals,

Hr~HPj, so

=

~rization. C = D.

Hence

c E

= D(x)

then

(d) ~ D,

H~P. jJ

nonzero.

of

(c)

= (c)(x)

by the uniqueness

so that

C ~ D.

and

for some ideal

D.

Since

Similarly,

each

(d)

of the lacD ~ C,

P. is a cancellation ideal. Suppose l with C ~ P.. Then c ~ P ~ H P = (x) i j_i J j J '

= D~P.

ideal,

If

= (d)(x)

P.

jJ

cellation

For non-

each

S

j~i J

necessarily

are the factorizations

respectively,

Hence

is an ideal of C ~P

d E D,

(e) = ~H~ = H P ~ = (d) k K

k ~ j

so

for some

(d) = H P ~ k K

Cx = Dx.

j=l J

that

C

so

is a can-

J

it follows

that

C = DP.

and hence that

P.

I

weak meet principal.

Then

P.

is

I

is b o t h weak meet p r i n c i p a l

and weak

I

join principal,

so

P.

is a p r i n c i p a l

element

of

~(S).

By T h e o r e m

I

2.3,

P.

is a strongly

principal

ideal of

S.

Hence

every p r o p e r

i

principal prime

ideal

of

S

is u n i q u e l y

a product

of strongly

principal

ideals. In some instances

an ideal w h i c h

principal

ideals b e h a v e s

principal

ideals.

as if it w e r e

146

is the finite finitely

union of join

generated

by strongly

ANDERSON/JOHNSON THEOREM

3.21

(Nakayama's

be an ideal of pal ideals ideals).

B

a. ~ AM,

i__~f A

is finitely

[AI,...,An]

a. E

be a semigroup

A,

generated

S

with

and

n

(a.)M

generate

S

and let

by strongly

A. = (a.) x I j. If

for

a contradiction.

principal

then

set of strongly

for some

A

set of join princi-

A ~ B U AM,

is a minimal

A = A IU -.-UA

then

~i,.-.,An

Let

is the union of a finite

is an ideal of

Assume

ideals with

Lemma).

which

(e.~., If

Proof.

S

A ~ B.

principal

i = l,--.,n.

i r j,

Hence

then

i = j,

If

AI,.-. so

(a.) l

= (a.)M.

But then

S = M U (0: a.)

1

so

a. = 0,

1

a contradiction.

1

Hence

a. E B for all i, so A ~ B. The proof of the more general i statement is somewhat similar and can be found in [3, T h e o r e m 1.4]. Some clarification

the following THEOREM

of T h e o r e m

3.21 is given by

theorem.

3.22

([3, T h e o r e m

join p r i n c i p a l ideals

of the hypothesis

1.5]).

ideal which

each of which has

If

S

is a semigroup

is the finite union

zero annihilator,

and

of strongly

then

J

J

is a

principal

i_~s strongly

prin-

cipal. SECTION Recall that an ment

a

satisfies

An equivalent

r-semigroup ab = a c r

statement

ideal

(a)

tinues

to denote

analog

implies in an

principal.

a semigroup

simple but interesting

is a semigroup

0

is that

is strongly

4.

of

section

M.

r-semigroups

ele~.

every p r i n c i p a l

this

ideal

every

for some unit

r-semigroup

Throughout

with maximal

property

in w h i c h

b = ~c

S

con-

We first note a

which has no good

in rings.

PROPOSITION

~.i.

S

G

and let

Let

S

be an

denote the group

r-semigroup. of units

of

Let S.

B

be an ideal

Then

B U G

of

is an

r-semigroup. Proof.

Clearly

elements

of

E G.

Hence

THEOREM

B U G

B U G

is a subsemigroup

with

B U C

ah = a c r

is an

([3, Lemma

~.8]).

every

ideal

A = (a,b)

that

~rime

ideals.

Then

M

of

then

S.

If

a,h

b = ~c

and

c

for some

r-semigroup.

4.2

~ower o f

0,

Let

S

be an

r-semigroup

is doubly generated

i_~s principal

in w h i c h

is a product o f

and every nonzero

ideal

is a

M.

Since ideals with

S

is an

r-semigroup,

AB = AC ~ 0,

then

if B = C.

147

A,B

and

C

We consider

are p r i n c i p a l the case of a

are

ANDERSON/JOHNSON semigroup THEOREM

in which all ideals satisfy this property.

4.3.

Let

cancellation

S

be a semigroup

property:

in which all ideals

AB = AC # 0

i__ssprincipal

implies

M2 = 0

o__rr M

Proof.

We borrow freely from the techniques

make two observations. AB ~ AC # 0, B ~ C.

then

Since

and every nonzero

Let

A,B

A(BUC)

and

= AC # 0,

(AUB)(AUB) implies

AB ~ A 2 U B 2.

Assume

M 2 # 0.

Let

x E M-P.

Then

M1~ = ( ( M - P ) U P ) P

(x) = (x) U P.

C

be ideals of

so

B U C = C

~ 0,

Now = 0,

so

so

M((x)UP)

It follows that

then

P = 0

(x)(y) ~ (x 2) U (y2) or

x E M-M 2.

Hence

If

M

~ M 3, 3.3).

Then

Let

M 2 ~ O,

x E M-M 2

s

S

(and

M

y 6 M,

since

M 2 # 0),

be the least positive

Hence

Now

M s = (x).

S,

is prime.

of

then 0

(Theorem

so

integer with

M s ~ Mx # 0 Since

implies

x E M-M 2,

it

Then

S

n

that

so

is nil-

S = (x).

r-semigroup, ).

Hence

If

fi (x n) # O, we get n=l ~ M n = 5 (x n) = O. In n=l n=l

from Lemma 3.14.

We will end this section with some results N-semigroups.

we have

n

sI E ~(x n=l

either ease the result now follows

r-semigroup.)

M

~ (x) n = (x) ~ (xn). For n=l n=l E S with y = s i x = s2x2

s

is an ~

the contradiction

is principal.

n

Then since

for each

M = (x)

we have shown that

then there exist

n

ian

so that

and

(y) ~ (x)

is the radical

we have shown that

0

fi (x) n, n=l

n

theory of

gives

M = (x).

s x .... .

s I E (x n)

N P

is prime), whence

M s = M s N (x) = (M s : x)x.

Suppose that

0 # y E

.....

M

is the only prime ideal of

potent. if

Also,

a contradiction.

Hence if M

and

xP = p2 = 0.

with Mx r 0, and choose n so that (M2U (x)) n = M 2 (M2U (x) )n-i with M 2 r M 2 U (x),

must be nilpotent.

follows that

so

M

x E M-M 2

Then

M s-I ~ (x),

from

M = (x).

M(M-M 2) r 0.

Choose

M s ~ (x).

If

(since

In either case,

is the only prime ideal of so

x n = O. M

(y) ~ (x).

and hence

= M(x) ~ 0

If

If

it follows that

P = P N (x) = P(x) = 0, S.

M.

We first

S.

xP ~ ((x 2 ) U P 2 )

(x) ~ (x 2) U P,

is the only nonzero prime ideal of

(x) ~ (y)

ideal is a power of

be a prime ideal different

((x) U p ) 3

= x2p U p2 = ( ( x 2 ) U p ) p . Then

P

satisfy the

Then either

used in [28].

2 = (AUB)(A 2UB2),

(AUB) 3 ~ 0

let

B = C.

(Recall that an

We begin by recalling

148

on the dimension

N-semigroup

is a Noether-

some facts from the

ANDERSON/JOHNSON dimension theory of local rings. Let

R

be a ring.

A prime ideal

exists a chain of prime ideals chain of length

n+2.

P

has rank

n

if there

P0 ~ P I ~ "'" ~ Pn = P'

but no such

Krull's Principal Ideal Theorem states that in

a Noetherian ring, a prime ideal

P

minimal over an ideal generated

by

n.

The converse of the Principal

n

elements has rank at most

Ideal Theorem is also true:

a prime ideal

P

of rank

Noetherian ring is m i n i m a l over an ideal generated by Suppose that dimension of

R

(R,M)

is a local

By the Principal Ideal Theorem, at least

d

elements.

ments, then

R

(Noetherian)

is defined to be rank

If

M

M.

n

in a

n

ring.

elements. The

Suppose that

any minimal basis for

M

must have

has a basis consisting of

is said to be regular.

are called a system o f parameters

if

Elements

(Krull)

d = dim R.

d

ele-

Xl,-..,x d

(Xl,...,x d)

is

of

M

M-primary.

By

the converse of the Principal Ideal Theorem every local ring has a system of parameters.

By the Principal Ideal T h e o r e m

est number of elements that can generate an be an

M-primary ideal and let

large

n,

degree

D(Q,n)

n

D*(Q,n)

=

D(Q,n)

where

local ring

R,

polynomial

for

R

is the few-

M-primary ideal.

be the length of

D*(Q,n)

with rational coefficients.

Hilbert-Samuel

d

is called the

with respect to

Q.

Hence for a

we have equality of the Krull dimension of

degree of the Hilbert-Samuel

polynomial.

Q

For

is a p o l y n o m i a l of

D*(Q,n)

fewest number of elements generating an

Let

R/Q n.

R,

the

M-primary ideal, and the For a proof of these facts,

the reader is referred to [15]. All three of these notions

can be defined for

more generally for Noether lattices. that for

N-semigroups,

or

As we shall see, it turns out

N-semigroups we still have equality of the Krull dimension

and the degree of the Hilbert-Samuel

polynomial.

However,

systems of

parameters may have length greater than the Krull dimension. We note that an ideals,

ating set for Let r,

N-semigroup

S

has only finitely many prime

since any prime ideal is generated by a subset of any gener-

S

denoted

M. be an

N-semigroup.

r a n k P = r,

A prime ideal

P

of

S

has rank

if there is a chain of prime ideals

~ PI ~ "'" ~ P = P but no such chain of length r+2. The Krull P0 T r dimension of S, denoted by dim S, is defined to be rank M. THEOREM 4.4.

Let

S

be an

N-semigroup.

149

Let

P

be a prime ideal

ANDERSON/JOHNSON o_~f S

minimal over an ideal ~enerated by

r

elements.

Then

rank

P~r. Proof.

Since

S

is an N-semigroup,

~(S)

is a Noether lattice.

The result now follows from the more general result for Noether lattices given by R. P. Dilworth Let S/Q n

Q

be an

[23, Theorem 6.5].

M-primary ideal of the

has finite length.

Let

D(Q,n)

N-semigroup

S.

be the length of

Then

S/Q n.

The

next theorem follows from the more general result for Noether lattices as developed in [32]. THEOREM 4.5. ideal.

Let

Let

S

be an

D(Q,n)

unique polynomial ents such that

N-semigroup and let

be the length o_~f S/Q n. D*(Q,n)

o_~f desre e

D*(Q,n) = D(Q,n)

Similarly,

if we define

minimal basis for

Mn,

is a polynomial of degree Thus in an

be an

with rational coefficin.

to be the number of elements in a

B(n) = B*(n)

dim S-I

for large

n

U [0]

(X)/(XY), so

F

S = F/(XY).

(Y)/(XY),

dim S = i.

and

Clearly

X S

(X,Y)/(XY)

and

Y,

is an

sup-

F = [xiy j I i,Ja 0]

N-semigroup.

Then

are the only prime ideals of

But since any principal ideal of (Y)/(XY),

However,

no principal ideal of

S,

S is contained in

(X)/(XY)

or

Hence

does not have a system of parameters of length

S

B*

with rational coefficients.

is the free semigroup on

and let

where

N-semigroup we have equality of the Krull dimension

and of the degree of the Hilbert-Samuel polynomials. pose that

M-primary

Then there exists a

dim S

for large

B(n)

then

Q

S

is

M-primary. i.

Our

next theorem will determine when systems of parameters of length dim S

exist.

THEOREM 4.6.

Let

prime ideal of

S

S.

be an Then

P

N-semigroup has at least

below it and there are exactly only i__ff P Proof.

and let r

We may assume that

r primes directl Y below

(Xl'''''Xs) ~ Qi"

r > 0.

P.

Hence

Let P

Let

QI,...,Qs

x i E P-Qi"

so

is minimal over

minimal over an ideal generated by

elements.

the

but that

P

is minimal over

Yi E P = Qj U Q~ for Qi's

~ ~ J,

don't contain all the

Yi'S.

150

P

if and

elements. be the prime

and if

(yl,-.-,yr). so

r

(Xl,.--,Xs).

s ~ r,

r

r

(Xl,...,x s) ~ P,

Principal Ideal Theorem (Theorem 4.4)

s > r,

be a rank

prime ideals directly

is minimal over an ideal ~enerated by

ideals directly below

then

P

Yi E Q~. Thus some

If

By the

s = r,

Finally,

but

P

is

suppose

Yi ~ QJ'

Thus at most

r

of

Qi m ( Y I " ' " Y r ) '

ANDERSON/JOHNSON

contradicting COROLLARY there

the fact that

4.7.

d

be the number

and let

d

be an

prime

ideals

(S,M)

(yl,...,yr). of dimension

directly b e l o w M

M

d.

Then

and there are

if and only i f

S

has a system

we define the following in a minimal

of elements

be the Krull

of

s > d.

of the inequalities

basis

generating

dimension

and that we may have

bilities

N-semigroup

of elements

number

over

d.

N-semigroup

be the smallest

is minimal

directly b e l o w

o f length

For an v

d

primes

o f ~arameters

d

(S,M)

are at least

exactly

Let

Let

P

S.

an

for

M,

let

M-primary

v m s

the various

In what

s

ideal,

We have seen that

We investigate

v ~ s a d.

numbers.

follows,

possi-

let

F V

be the free m o n o i d on XI,...,X v with n n F v = [XII...X v v I nl' 9 "''nv ~ 0~ U {0~. be the ideal of

F

generated

0

adjoined,

Also let

by all products

so

X(r) '

of

r

i ~ r ~ v'

distinct

basis

V

elements

XI,..-,X v.

(We may extend our definition

X(v+l)

= 0.)

Hence

X(1) = (XI,...,Xv) ,

while

X(v) = (Xl'''Xv)" Suppose we set

S = F /X(1) 2

Then

V'

course m a y be arbitrary. then

v = s > d.

to

the maximal

X(O)

= S

ideal of

v ~ s = d = 0

and

and Fv ,

v

of

9

If we let

If we take

S = F /X(d+l) v then S = F

d = v,

where so

v > d,

v = s = d

V

and

S

Then

is regular.

Finally,

v ~ s = v-i ~ d A local Noether

a join of a unique usually

dim~

where

principal

denoted by

indeterminates

(~,M)

over

K.

U ~ l~xd+l'''" 'vXd+l]~)"

is said to be regular

elements 9

regular

RLn.

S = Fv/(X(d+l)

0 ~ i ~ v-d.

lattice

distributive

let

It is known

local Noether

Let

K

Then

M

is

[18] that there

lattice

of dimension

be a field and let

RL

if

XI,...,X n

is the sublattice

is n--

be

of

n

~(K[XI,...,Xn])

generated

(XI),...,(X). n the free monoid S

is regular

under joins

It is also known on

n

[i] that RL

n It follows

generators.

of dimension

n

and products

of the ideals

~ ~(F

if and only if

) where F is n n that an N-semigroup

~(S) ~ ( F

). n

Given x ~ y an

any semigroup

if and only if

r-semigroup

r-semigroup, tion:

or an

then

if and only if semigroup.

Then

satisfies

implies

S/~ S

w~ can define the congruence Clearly

N-semigroup,

S/~

ab = ac # 0

S,

(x) = (y).

then so is

the stronger

b = c.)

is regular. is regular

Let

An S

S/~.

be an

If

(If

S

cancellation

N-semigroup

if and only if

151

~

~(S) ~ ( S / ~ ) .

S

n-dime~ S/~

by S

is

is an condi-

is regular ial

N-

is ~ o m o r p h l c

to

ANDERSON/JOHNSON

F

where

n = dim S.

n

A deep theorem from Noetherian rings says that a regular local ring is a UFD [35, T h e o r e m 184].

An

r-lattice

UFD if every principal element can be written as a product of principal primes.

domain is called a

(necessarily uniquely)

It remains an open question

whether a regular local Noether lattice is a UFD.

However,

it is

easily seen [i] that a distributive Noether lattice domain UFD if and only if

~

is regular

(i.e.,

~

Noether lattice for each maximal element

M

have shown the following related result. tice domain

~

the following conditions

~

is a

is a regular local of

~).

The authors

For a quasi-local are equivalent:

[9]

r-lat-

(I) ~ R L

, n

(2)

~ is a regular local Noether lattice with exactly

prime elements,

and (3)

~

is a UFD with exactly

n

n

rank one

nonzero princi-

pal primes. Let of

Fn

ideals

Fn

be the free m o n o i d on

with

Xl,...,X n.

Let

A ~ (XI...Xn) = ( X I ) N ... n (Xn)"

(XI)/A"'''n(X)/A

A

be an ideal

Then in

Pn/A

the

are principal prime ideals and each prim-

cipal ideal of An

F /A is a product of these principal prime ideals. n r-semigroup S will he called a ~ - s e m i g r o u p if every principal

ideal of

S

~-semigroup

is a product of prime ideals.

It can be shown that in a

every principal ideal is actually a product of principal

prime ideals.

Thus a

~-semigroup

domain is actually a UFD semigroup

in the sense that every principal ideal m a y be written u n i q u e l y as a product of principal prime ideals. THEOREM 4.8. group with

n

The free semigroup

Fn

__~

principal primes.

If

S

principal primes com~ruence Let

then

x ~ y S

: (•

is a UFD semi-

is isomorphic

to

F

(where

~

n

is the

if and only i_~f (x) = (y)).

be a

w - s e m i g r o u p w h i c h is not a domain.

finite number of rank morphic __t~ Fn/A

S~

X I,-'',X n

is a UFD semigroup w i t h

0

where

prime ideals, say A

is an ideal of

n. Fn

Then

Then

$/~

with

S

has a

is iso-

A ~ (XI-..Xn)

(Xn)"

Proof.

This is the semigroup v e r s i o n of the result for lattices

given in [12]. Let F

F

be the free

(commutative)

semigroup on a set

is a UFD semigroup w i t h principal primes

number of principal primes is semigroup with

~

(~

IX I .

any cardinal)

152

[(x) ] x E X].

Conversely,

if

S

X.

Then

Here the

is a UFD

principal primes, then

S/~

is

ANDERSON~JOHNSON

isomoprhic

to

Let

F

the free semigroup

M = (XI,-.-,X r)

on a set with

be the m a x i m a l

~

ideal of

elements.

F

the free r'

semigroup on

X I , . - . , X r.

Then

Mn

n

all elements each

of the form

n.l ~ 0.

elements.

Thus

Mn

T H E O R E M h.9.

Let ideal

erators w i t h Noetherian

S

elements.

Conversely,

n,

Mn

S

as our last theorem. of Krull dimension

has a m i n i m a l

S

r

set of @en-

suppose that

Assume that

has a m i n i m a l

with

(n+r-l~ \ r-i ;

S

is a

has d i m e n s i o n

r

set of 6 e n e r a t o r s w i t h

is a regular

N-semigroup

and hence

S/~

r

statement

follows

from the remarks

of the previous

To prove the second statement we only need show that

generated by

r = S

let

Then

-

A ~ B

of

J = 0.

Mn

of

F .

r-semigroup,

since

minimal

N-semi@roup

{n+r-l~ \ r-i ;

The first

paragraph. is an

be a r e 6 u l a r

= n

c o n s i s t i n g of

and its converse

n,

to -

Proof.

result

For every

(

- -

n I + --. + n r

has a m i n i m a l basis

semigroup w i t h

~s isomorphic

where

M.

and that for every n+r-l~ r-I / elements.

consisting

n

XII.--Xr r

We state this

with m a x i m a l

has a m i n i m a l basis

is N o e t h e r i a n

set of generators

al,.-.,a r

S

for then S is an N - s e m i g r o u p and M can be r u elements. Recall that A ~ B AB implies r-i )

of degree

for n

~ = 0.

and M.

al, -. .,a r

Let

be a

We claim that the power products

form a m i n i m a l base

for

M n.

If not,

n

be the least integer for which this is false 9 Now M n has a /n+r-l) minimal base with ~ r-i elements and every element of M m is a unit m u l t i p l e

Hence, the m i n i m a l b a s e of a power product

Mm

of degree

lie in a m i n i m a l basis n.

al, 9 .. ,a r

with

'n+r-l~ \ r-i I

s < n.

for

Ms ,

Hence the power products

m i n i m a l b a s e for al,-..,a

of

of a power product

M n,

of different

degrees

elements must

contain

But then this power product w h i c h contradicts

of degree

for all

of some degree.

n.

n

are distinct

Moreover,

of

and form a

power products

are distinct.

cannot

the m i n i m a l i t y

of

Hence the power prod-

r

ucts of

al, 9 ..,a r

satisfy cancellation, SECTION

In this b r i e f

so

S

is an

5.

section we give an o v e r v i e w of the literature.

The number of papers

on semigroups,

is vast as m a y be seen by consulting

even c o m m u t a t i v e

Clifford and P r e s t o n

We have m a d e no attempt to give a complete b i b l i o g r a p h y tive semigroups.

r-semigroup.

R a t h e r we have given some papers

ideal t h e o r y of c o m m u t a t i v e

semigroups

153

semigroups, [21-22].

of commuta-

c o n c e r n i n g the

that resemble portions

of

ANDERSON/JOHNSON

"multiplicative ideal theory" in commutative rings.

As Dedekind do-

mains and Prefer domains play a central role in multiplicative ideal theory, it is not surprising that various of the many equivalent conditions characterizing Dedekind domains or Prefer domains have been investigated for commutative semigroups, especially for cancellation monoids.

Such papers include [24-26],

[31], and [46].

tion semigroups have received attention [43-45].

Multiplica-

The papers [50] and

[52-56] look at ideal theory in a commutative semigroup. One of the earliest papers on multiplicative lattice theory is given by Krull [36].

An early systematic study of multiplicative

lattices was given by Ward and Dilworth [58]. weak meet principal element was introduced. tributive multiplicative lattices.

Here the notion of a Ward [57] studied dis-

In [23], Dilworth introduced the

notion of principal element and defined a Noether lattice. Anderson defined

r-lattices.

In [3],

Other papers on multiplicative lattice

theory include [i], [4], [7-12], [18-19], and [32-33]. Aubert introduced the theory of

x-ideals in [17].

Aubert's

x-ideals subsumed the earlier work on ideal systems given by Jaffard [30], Lorenzen [40-42], and Prefer [49].

Papers concerning

x-sys-

tems, other than by Aubert and his students include Johnson and Lediaev [34] and Lediaev [38] which compares Noether lattices and

x-

systems. The early papers on ideal theory in commutative semigroups, Arnold [14], Clifford [20], and Ward and Dilworth [59], are mainly concerned with divisorial ideals, probably due to the recent introduction of divisorial ideals into commutative ring theory.

However,

in [59] it was observed that the lattice of ideals of a semigroup forms a multiplicative lattice, that the principal ideals in a semigroup are weak meet principal, and that the general results from multiplicative lattices show that every ideal in a Noetherian semigroup has a primary decomposition.

The next important paper on ideal

theory in commutative semigroups was given by Aubert [16]. paper served as a prelude to his work on

154

x-ideals.

This

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9.

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i0.

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iI.

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30.

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33.

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Department of Mathematics The University of Iowa lowa City, Iowa 52242 Received October 17, 1983 and February

9, 1984 in final form.

158