A simple proof of an extension of Faith's correspondence theorem for projective modules is given for a Morita context (R, V, TV, S) in which VWV = V and WVW ...
Period&x
ON FAITH’S
Mathemuticu
Hu7tgariuz
VoZ. 21 (I),
CORRESPONDENCE W. K. NICHOLSON
(1990),
pp. 31-34
THEOREM
(Calgary)
Abstract modules
A simple proof of an extension of Faith’s correspondence theorem for projective is given for a Morita context (R, V, TV, S) in which VWV = V and WVW = W.
In 1971, Carl Faith [2] announced his correspondence theorem for projective modules, a result which has proved to be very useful in the study of noetherian simple rings (see Cozzens-Faith [l] and Faith [3], page 195). In this note we present a simple proof of an extension of Faith’s theorem. Throughout the paper R and S will denote rings with unity and all modules will be unitary. If $V’, is a bimodule let lat (aWR) denote the lattice of all sub-bimodules of W, with similar notations for left and right modules. If A c R and B E S are idempotent ideals (that is A2 = A and B2 = B) define B * lat (aW) = { BX ) X E lat (aW)} = {X E lat (sW) 1BX = X} lat ( WR) . A = { XA / X E lat ( WR)} = {X 6 lat ( W,) 1XA = X} B * lat @V,) * A = { BXA )X E lat (sWR)} = {X E lat (sWR) ) BXA = X}. LEMMA. Using the above notation: (I) B * lat (SW) is a lattice via X, V X2 = Xl + X, and X, A X, = B(X, I-I X2,. (II) lat (WR) . A is a lattice via Xi V X, = X, + X2 and Xl A X2 = (Xl n X,)A. (III) B * lat tSWR) - A is a lattice via X, V X, = X, + X2 and
&AX, = B(X,n X&t. PROOF. In each case X1 + X, lies in the set in question and so is the join of X1 and X,. For the rest it suffices to do it for (III) since we can take A = R or B = S. We have B(X, n X&A E B. let (@Vs) * A because A2 = A and B2 = B, and clearly B(X, n X&A _C BX,A n BX,A = Xi n X,. But if
16A30.
Research supported by N.S.E.R.C. (Canada), Grant No. A 80’76. Mathematics Buaject clusaiftiion numbers, 1980/86. Primary 16A60; Key word8 and phraae8.
Morita
contexts,
projective
Secondary
modules. Budoiw#
ElulG.3
Dordf&t
32
NICHOLSON:
ON FAITH’S
CORRESPONDENCE
THEOREM
Y c X1 n X2 and Y C B - lat $$VS) . A then Y = BYA c B(X, n X,)A. Hence B(X, n X2) A is indeed the meet of X, and X,. A Morita context (R, V, W, S) consists of two rings R and 8, two bimodules RVs and s WR, and two products V X W - R and W X V -+ 8, written multiplicatively,
such that the “matrices”
RV
form a ring with the usual [ WS I matrix operations. This amounts to insisting that the products induce bimodule homomorphisms V @ ,W -+ R and W @ RV + ASand that (vw)q = v(wvl) and (wv)q = w(wq) hold for all 21,v, E V and w, w, 6 W. The ideals VW in R and WV in S are called the trace ideals of the context.
THEOREM. Let (R, V, W, S) be a Morita context, write A = VW B = WV, andassume VWV = V and WVW = W. Then A2 = A and P and the following are lattice isomorphisms: (I) ‘p: A * lat (qR) - B * lat (sW) where Lg, = WL; and Xv-1 = Moreover q induces a lattice isomorphism A - lat (*RR) -+ B * lat (s WR). (II) y: lat (RR) . A + lat ( Vs) . B where Ty = TV and Xv-1 = Moreover y induces a lattice isomorphism lat (RRR) - A -+ lat (RVs) * B. (III) 5: A * Iat (RRR) . A + B . lat (sSs) * B where 15 = WIV; monoids. JE-1 = VJW. Moreover 5 is an isomorphism of multiplicative
and = B VX. XW’. and
PROOF.
(I). First, Lg, = WL lies in B - lat (sW) since B(WL) = (WV)WL = = WL. Similarly, VX E A - lat (RR). Now L ---f WL ---r V( WL) = AL = L and X --f VX -+ W( VX) = BX = X show these maps are inverse bijections. They clearly preserve inclusions and (L + L,)cp = W(L + Ll) = WL + WL, shows cp preserves joins. As to meets, observe that L~J A Ll’p = B(Lv n L,q)
= WV(WL
(L AL& = [A(L n L,)lg, = WA(L n LJ =
n WL,)
WVW(L
n L,).
The fact that W(L n L,) c WL n WL, shows WVW(L n Ll) E WV( WL n n WLl). For the reverse inclusion, let w E WL n WL,. Then VW 5 V WL n n VWL, EL n L,, so that WVW E W(L n Ll). Hence WV(WL n WL,) E 5 W(L n Ll) = W VW(L n Ll). This shows 9 preserves meets, and so proves the first statement in (I). Then the second statement follows since both cp and 9-l induce the maps indicated. (II) Analogous to (I). = V, (III) We have WIV E B slat ($s) * Bsince WVW = W andVWV andsimilarly VJW c A *lat (RRR) * A.ThenI -+ WIV - VWIVW = AIA = ---f WVJW V = BJB = J show the maps are inverse bi=landJ+VJW
NICHOLSON:
jections, = WIV
ON FAITH’S
CORRESPONDENCE
33
THEOREM
and they are clearly order preserving. The fact that + W&V shows 5 preserves joins. We have
W(I + I,) V =
(0 I,)E= [&In ~lME = WVW(1n I,)VWV= W(I n I,)V, 15AIIE = B(IEn 1,4)B= wv(wrV n WI,V)WV. Then the fact that W(I n IJP E WIP n W1,V shows (I A Ii)5 _c 15 A 115. On the other hand, ifs E (VU n WI,V) then VsW E TTWITrW n VvW1,VW E c 1 n Ii so 15 A 15, E W(I n 1JV = (1 A I,)[. Thus 5 is a lattice isomorphism. Finally II, = IAI,so (11,)5 - W(IAI,)F’ = (WlV)(WI,P) = (1Q(112). This, together with A5 = VA W = B3 = B, shows 5 is a monoid morphism. Call (R, V, W, S) a corrupondence context if VW V = V and WV W = W. One instance of this is when WV = S, equivalently when RV is finitely generated and projective (and W, is a generator). Specializing the theorem to this case gives Correspondence Theorem Let *V be a finitely generated projective W = Vd = horn R( V, R). Then (R, V, W, WV = S. Writing A = VW as before, we COROLLARY.
for Projective Modules - [Faith 21. module and write 1.9= end RV and S) is a corr@pondence contest with have the lattice isomorphisms:
(I) ‘p: A * lat ($2) + lat (s W) with Lg, = WL, Xv-l = VX; induces a lattice isomorphism A * lat (RRR) - lat (s WR). (II) y: lat (RR) * A - lat (V,) with TV = TV, Xv--l = XW; induces a lattice isomorphism lat (&s) A - lat (RVs). (III) 5: A . lat (RBR) * A -+ lat ($,) with 15 = WIV, J5-l= of multiplicative monoids. and 5 is an &morphism
and y and
p
VJW;
One example of a correspondence context where WV f S is the following: Let A be any proper ideal of a ring R such that A2 = A, and let S be the subring of R generated by A and the unity. Then (R, A, A, S) is a correspondence context and WV = A2 = A # S. In particular this is true of any proper ideal A if R is regular in the sense of von Neumann. Similarly, (R, A, A, A) is a correspondence context whenever A = Re where e2 = e is a central idempotent in 22. The situation in the corollary can be generalized. If RM is any module and we write Md = horn R(M, R) and S = end R(M), then (R, M, Md, S) is called the derived context for M. This is a correspondence context if Md M = S (that is, RM is finitely generated and projective - the situation in the corollary) or if MMd = R (RM is a generator). Zimmermann-Huisgen [5] calls RM locally projective if, given the diagram with exact row and 3 Periodica
Math. 21(l)
34
NICHOLSON: ON FAITH'S CORRESPONDENCE THEOREM
/. {%,m2,“.,
mk} E Af, there exieta
4 J
AA
I
p
B-o
y: iv-+ A such that m, ya = mib for each i = 1,2, . . . , k. She shows (among other things) that M = NMdM in the derived context. If, in addition, i@ is locally projective ae a right R-module, we get MdNMd = Md, and so (R, N, i%Zd,S) is a correspondence context and the conclusions of the corollary hold. One situation in which this happens in if JK is regular (Zelmanowitz [4]) that is for all m E M there exits 1 C Nd such that m = (mA)m. If we take p = A(mA) then p E Md and we have m = (mp)m and ,U = p(m,u). Hence (R, M, Md, 8) is a correspondence context for each regular module. REFERENCES [l]
J. COZZENS and C. FAITH, Simple Noetherian rings, Cambridge University Press, 1976. ZbZ 314: 16001 [S] C. FAITH, A Correspondence theorem for projective modules and the structure of simple no&h&m rings, BulL A. M. S. 77 (3)? 1971, 338-342. MR 431 6264 [3] C. FAITH, Algebra: Rings, modules and categwws, I, Springer, New York, 1973. MR 51: 3206 Regular Modules, Trans. A. M. S. 163 (1972), 341-356. MR [4] J. yi, yo;~~z,
[6] B. Z-N-HUISGEN, Pure Submodules of Direct Math. Ann. 224 (1976), 233-246. 262 3218 16022 (Received March DEPARTMENT OF MATH. A%D STAT. UNlVERSITY OF CALGARY CALOARY,CANADATSN lN4
1, 1988)
Products
of Free Modules,
