NEW ZEALAND JOURNAL OF MATHEMATICS Volume 31 (2002), 115-129
PROJECTIVE, FLAT AND MULTIPLICATION MODULES
M a j i d M . A l i a n d D a v i d J. S m i t h (Received December 2001)
Abstract. In this note all rings are commutative rings with identity and all modules are unital. We consider the behaviour of projective, flat and multipli cation modules under sums and tensor products. In particular, we prove that the tensor product of two multiplication modules is a multiplication module and under a certain condition the tensor product of a multiplication mod ule with a projective (resp. flat) module is a projective (resp. flat) module. W e investigate a theorem of P.F. Smith concerning the sum of multiplication modules and give a sufficient condition on a sum of a collection of modules to ensure that all these submodules are multiplication. W e then apply our result to give an alternative proof of a result of E l-B ast and Smith on external direct sums of multiplication modules.
Introduction
Let R be a commutative ring with identity and M a unital i?-module. Then M is called a multiplication module if every submodule N of M has the form IM for some ideal I of R [5]. Recall that an ideal A of R is called a multiplication ideal if for each ideal B of R with B C A there exists an ideal C of R such that B — AC. Thus multiplication ideals are multiplication modules. In particular, invertible ideals of R are multiplication modules. On the other hand cyclic modules are multiplication, and multiplication modules over local rings are cyclic, [4, Theorem 1] and [2, Theorem 2.1]. Let R be a ring and K and L submodules of an /2-module M. The residual of K by L is [K : L], the set of all x in R such that xL C K. The annihilator of M, denoted by ann(M), is [0 : M], and for any m & M the annihilator of m, denoted by ann(m), is [0 : Rm\. For an i?-module M, we put 6 (M ) = • ^1- Let M be a multiplication module and N a submodule of M. There exists an ideal I of R such that N = IM. Note that I C [N : M] and hence N = IM C [N : M ]M C iV, so that N = [N : M]M. Note also that K is a multiplication submodule of M if and only if N D K = [N : K] K for all submodules N of M. In Section 2, we investigate the tensor product of modules. We show that the tensor product of two multiplication i2-modules is a multiplication i?-module, The orem 2.1. We also prove that if M is a multiplication i?-module and P is a projec tive (resp. flat) i?-module such that ann(M) C ann(P), then M ® P is a projective (resp. flat) A-module, Theorems 2.3 and 2.5. We apply Theorem 2.3 to gener alize [18, Theorem 11] and show that multiplication modules whose annihilators are generated by an idempotent are in fact projective modules, Corollary 2.4. As
2000 A M S Mathematics Subject Classification: 13C13, 13A15, 15A69. K ey words and phrases: projective module, flat module, multiplication module, free module, ten sor product, direct sum. 116 MAJID M. ALI AND DAVID J. SMITH a consequence of Theorem 2.5, we give a generalization of [16, Theorem 2.2] and prove that if M is a multiplication i?-module such that ann(M) is a pure ideal, then M is a flat i?-module, Corollary 2.7. Section 3 is concerned with the sum of modules. If N\(X G A) is a collection of submodules of an P-module M such that the sum of any two distinct members is multiplication, then need not be multiplication [18]. Theorem 3.1 gives several equivalent conditions under which this sum is multiplication, generalizing P.F. Smith’s theorems [18, Theorems 2 and 8(i)]. We also consider the following question: Let R be a ring and N\(X G A) a collection of submodules of an i?-module M such that *s multiplication. Is at least one N\ a multiplication module? We show that the answer is no, and we give a sufficient condition on such a collection to ensure that all of these submodules are multiplication, Theorem 3.2. We apply Theorem 3.2 to give an alternative proof to a result of El-Bast and Smith [ 8 , Theorem 2.2]. All rings are commutative with identity and all modules are unital. For the basic concepts used, we refer the reader to [6], [9], [10], [12] and [13].
1. Preliminaries P.F. Smith gave necessary and sufficient conditions for the sum of a collection of multiplication modules to be a multiplication module [18, Theorem 2]. We inves tigate this theorem and prove a strengthened version using alternative methods.
Lemma 1.1. Let R be a ring and N\(A G A) a collection of submodules of an R-module M, let N = X^AeA and let A = ^2\eA[N\ : N]. Consider the following statements: (i) N is a multiplication module.
(ii) H = A H for all submodules H of N .
(iii) R = A + ann(n) for all n G N.
(iv) [N : K] + ann(n) = X^agA^* : + ann(n) for all submodules K of M and all n G N.
(v) K H N — J^AeA K ^ A^a for all submodules K of M .
Then (i) => (ii) <^> (iii) <^> (iv) => (v). If all N\ are multiplication modules, then the first four conditions are equivalent.
Proof, (i) => (ii). As N is multiplication, N\ = [N\ : N]N for all A G A, and hence N = A N . Let H be any submodule of N. Then H = IN for some ideal I of R , and hence H = I N = I (AN ) = A(IN) = AH.
(ii) => (iii). Let n G N. Then Rn = An, and hence R = A + ann(n). (iii) => (ii) is obvious.
(iii) => (iv). Let K be any submodule of M. Let n G N. Clearly
^^[Ar> : K] + ann(n) C [N : K] + ann(n).
a Conversely, there exist a finite subset A' of A and elements x\ G [N\ : N](A G A') and z G ann(n) such that 1 = J^xeA' x A + w G [N : K] + ann(n). Then w = wi + W2 where W\K C N and W2 G ann(n), and hence w ~ wix x + [w\z + W2) G : K] -I- ann(n) AeA' AeA so that (iv) is satisfied, (iv) =>■ (iii) is clear by taking K = N. To complete the proof of the first assertion of the Lemma, it is enough to show that (iii) => (v). We do this locally. Thus we may assume that R is a local ring. If N = 0, the result is trivial. Thus we may suppose that N ^ 0. There exists 0 ^ n G N so that ann(n) 7^ R , and hence A = R. There exists fi G A such that [N^ : N] = R and hence N = N^. This implies that KHN = KnN^Q^KnNxCKnN, AeA so that K fl N = X^AeA N\K fl N\, and the proof of the first part of the lemma is complete. For the last assertion, it is enough to show that (ii)=>- (i). We first prove that N\ = [iV* : N]N for all A G A. We do this locally. Suppose that P is a maximal ideal of R. We discuss two cases. Case 1: A C P. Then for all n G N , Rn = An C Pn, and hence (Rn)p = P p(R n)p, and by Nakayama’s Lemma we have that ( Rn)p = Op, It follows that Np = Op, and hence (N\)p = Op for all A G A. Both sides of the equality N\ = [N\ : N ]N collapse to Op. Case 2: A <£. P. There exists /i G A and p G P such that (1 — p)N C N^. It follows that Np = (NfL)p, and hence for all A G A, [Nx-.N^p C [N\ : (1 — p)N]p = [[JV\:jV]:fl(l-p)]p = [[ATaN]p : : (R(l - p))P] = [iVAN]P : Q {Nx -.NJp , so that [N\ : N^\p = [N\ : N ]p. This implies that [Nx :N ]p N p = [N x -.N ^ p N p = [Nx:Nf,]p(NfM)p = (NxnN^)p = {N\)p n {Nfj,)p = (N\)P f l Np = (N\)p. From the two cases, we infer that the equation N\ = [N\ : N ]N is true locally and hence globally. Now assume that K is a submodule of M. Then by (v) of the lemma, KnN = Y.K nN x = ’£[K :NX]N,, AeA AeA = : N *\iN * : N]N C [K : N}N C K n N, AeA so that A 'fliV = [ K : N ]N , and hence N is a multiplication module. □ 118 MAJID M. ALI AND DAVID J. SMITH Note that the implication (v) =^> (iii) in the above Lemma is not true. Let R be a Priifer domain but not Noetherian. For example, take any non Noetherian valuation domain. There is a maximal ideal P of R which is not finitely generated. P is not multiplication, and hence 0(P) ^ R , [2, Theorem 2.1] and [3, Theorem 1]. However, for every non-zero ideal / of R, i n P = ^2peP In R p . The same example shows that (iii) does not imply (i), by taking Ni = N 2 = P. The following consequence is straightforward. Corollary 1.2. Let R be a ring and M an R-module. Then M is multiplication if and only if 9{M ) + ann(ra) = R for all m G M . Consequently, M is multiplication if and only if Rm = 9(M)m for all m G M. Let M be an .R-module and let S = {ra,\|A G A} be a set of generators for M. Let F be the free i?-module generated by the set S' — {xa|A G A} and let n : F —> M be defined on S' by 7t(xa) = m\ and then extended to F linearly. Let K be the kernel of ir. Then the following exact sequence is called a presentation for M: 0 — » K — > F ^ M — > 0. The following characterizations of projective and flat modules can be found in [7] and [15]. Lemma 1.3. Let M an R-module. Then M is projective if and only if for each presentation 0 — > K —> F M —>0 for M , there exists an R-homomorphism 8 : F —> F such that (i) 7T o 9 = 7T and (ii) ker0 = ker7r. Lemma 1.4. Let M be an R-module. Then M is flat if and only if for each presentation 0 — > K — > F ^ M — >0 and each u G K , there exists an R-homomorphism 9U : F —> F such that (i) 7T o 9U — tv and (ii) 9u{u) = 0. We also need the next result about flat modules. The proof can be found in [7]. Lemma 1.5. Let M be aflat R-module with a presentation as above. Let Ui G K, 1 < i < n and let 9i be the corresponding homomorphisms given above. Then there exists a homomorphism 9 : F —» F such that (i) 7T o 9 = 7r and (ii) 9(ui) = 0 for all 1 < i < n. 2. Tensor P roducts Let R be a ring. It is well known that if I and J are projective (resp. flat, multiplication) ideals of R , then IJ is projective (resp. flat, multiplication), [3] and [15]. If M is a multiplication .R-module and I is a multiplication ideal, then IM is a multiplication .R-module, [ 8]. If M is a finitely generated multiplication .R-module and I is a projective (resp. flat) ideal of R such that ann(M )fl/ = 0, then IM is a projective (resp. flat) .R-module [14]. It is also well-known [ 6 ] that the tensor product of two projective (resp. flat) i?-modules is a projective (resp. flat) .R-module. We now prove that the tensor product of two multiplication .R-modules is a multiplication .R-module. We also show that if M is a multiplication .R-module PROJECTIVE, FLAT AND MULTIPLICATION MODULES 119 and N is a projective (resp. flat) .R-module such that ann M C ann N, then the tensor product M <8 N is a projective (resp. flat) i?-module. Theorem 2.1. Let R be a ring and M i and M 2 multiplication R-modules. Then M i (8 M 2 is a multiplication R-module. P roof. We first prove that 9(M i)9(M 2) C 9(M i <8 M2). Let mi E Mi and m2 £ M 2. If x G [i?mi : Mi] and y G [i?m2 : M2], then xM i C i?mi and y M 2 C flm 2. It follows that xy(M i <8 M 2) = xM i <8 y M 2 C Rrrii <8 Rm 2 = i?(mi <8 m2), and hence x y G [i?(mi (8) m2) : Mi R(mi (8 m 2) = Rrrii <8 Rm 2 = 0(Mi)mi <8 9 (M 2)m 2 — 9(Mi)9(M2){mi <8 m2) C 0(Mi <8 M2)(mi <8 m2) so that R(mi (8 m2) = 0(Mi <8 M 2)(mi <8 m2). From this we obtain that Rx = 9(M i ® M 2)x for all x G Mi <8 M2, and the result follows by Corollary 1.2. □ The next lemma should be compared with [16, Lemma 3.6]. Lemma 2.2. Let R be a ring and M a multiplication R-module. Let {m A|A G A} be a set of generators for M . Then for each A G A, there exist a finite subset Aa of A and elements t^a G R with fi G Aa (depending on X) and a G A such that the following conditions are satisfied: (i) rnx = Y, V I \neA\ J (ii) t^ m p = t^pma for all a, ft G A and all fi E A\. (iii) mA = ^ P roof. Let A G A. It follows by Corollary 1.2 that 9(M ) + ann(mA) = R • There exists a finite subset A* of A such that ^ [Rmp : M] + ann(mA) = R- It follows by [16, Lemma 3.5] that there exist G [Rm M : M], d^, d\ E R and w\ E ann(mA) such that d» Cl» + dxW A = L As G [Rmn : M], cmmM C i?mM, it follows that for all a E A there exists P such that — c^Q.m.^. Put t — d^c^^c^d. Then E dvcln mA + dxw 2xm\ = mA, (mgaa / 120 MAJID M. ALI AND DAVID J. SMITH and hence m\ = ( ^ MeAAW j mA- This proves (i). Let cn,/? G A and E A\. Then 3 — d ^C ^^C fl — d u p 0^0,171 n t[j,f3Tna -— d^C^^C^pTYlf^ — d^C^f^C^pTTl^ and hence (ii) is satisfied. By (i) and (ii), it is now clear that m x = ^ t ^ m x = ^ 2 t^xrrifj,. □ Theorem 2.3. Let R be a ring and M a multiplication R-module. If P is a projective R-module such that ann M C ann P, then M P roof. Let {ra,\|A G A} be a set of generators for M. Let F\ be the free .R-module generated by {xa|A G A} and define 7Ti : F\ —» M by 7Ti(xa) = m\ for all A G A. Define 9\ : F\ —> F\ by 9\{x\) = J2u£Ax where A\ and t^\ are as defined in Lemma 2.2. It follows that for all A G A, (ttl o 91 )(xx) = 7ri(6>i(arA)) = 7Ti I t^\x^ = ^ \neAx ) fJ-& aa = y :tfiXm/i = mx = 7ri(rcA). /x€:Aa Hence 7Ti o = 7Ti . Assume that P is generated by {pk\k G /} . Let P2 be the free P module gener ated by {yk\k G I}. Consider the exact sequence 0 — * K 2 — ► P2 P — + 0, where 7r2 : P2 —» P is defined by 7r2(^fc) = Pk f°r all k E I, and iC 2 = ker7r2. It follows by Lemma 1.3 that there exists an P homomorphism 92 : P2 —> P2 such that (i) 7r2 o 92 = 7r2 and (ii) ker 7r2 = ker#2. The P-module M 0 P is generated by the set {m^ <8>Pfc|A G A, A: G / } . Let F = F\ ® F2 and consider the exact sequence 0 — > K — > F M 0 P — >0, where 7r = 7Ti ® 7r2. Define 9 : F —> F by 9 = 9\ 0 02. Then 7t(xa <8> yk) — tti(^a) ® 7T2(2/fc) and 9(x\ (7r o 9)(xx Case 1: y E ker it2 - Then y E ker 02, and hence # 2(2/) = 0- It follows that 0(ar ® y) = 6*i (rc) ® 0 = 0, and hence x ® i/E ker 0. Case 2: ar E ker 7Ti. There exists a finite subset A' of A such that x — X^aeA' wax a with wa E R. Hence 0 = 7Ti(ar) = 7Ti I ^ 2 W°cXoc I = ^ 2 Wa7rl(Xa) = ^ Wam a- VaG A' / a£A' a£A' Next, 0(x S ® “ S ® 02(y ) ( * ) ^aGA' /i£Aa J where Aa and t^a are as defined in Lemma 2.2. Since YlaeA' we have for all (3 E A, I X I W* tw m l3 = ^ 2 Wat ^ m a = 0. VaG A' / aGA' q £A' Hence XlaeA' £ ann(m^), and therefore X)«gA' ^t^awa £ annM C ann P. It follows that tn 0 = 02 I X t^away j = X t^awa62(y). VaG A' / aGA' Hence from (*) we have that 6 (x Although not all projective /2-modules are multiplication, it is well known [19] that every projective ideal is multiplication. Moreover, it is proved [15] that if I is a projective ideal then ann I = ann L for some idempotent ideal L of R. More is known for the finitely generated case. For example, it is shown [16] that a finitely generated ideal / of R is projective if and only if I is multiplication and ann I = Re for some idempotent element e. This result has been extended for modules [18]: If M is a finitely generated multiplication A-module such that ann M = Re for some idempotent e, then M is projective. The next consequence of Theorem 2.3 is a generalization of this last result to multiplication modules (not necessarily finitely generated). Corollary 2.4. Let R be a ring and M a multiplication R-module such that ann M = Re for some idempotent e. Then M is projective. 122 MAJID M. ALI AND DAVID J. SMITH Proof. As ann M = Re = ann(l — e) and R( 1 — e) is a projective ideal of R , we infer from Theorem 2.3 that M 0 R( 1 — e) is a projective i? module. But R = Re-\- R( 1 — e). Thus M ® R = M ®R e + M Finally M ® Re = eM ®R = 0<8>R = 0, and M Theorem 2.5. Let R be a ring and M a multiplication R-module. If P is a flat R-module such that ann M C ann P , then M Proof. Assume that Fi, F 2 and F are as in the proof of Theorem 2.3. Also assume that 7Ti, 7t2, 7t and 9\ are as defined before. Let x ® y be a generator in ker 7r. To show that M eg) P is flat, we need to show that there exists an P-homomorphism '• F —> F such that (i) 7r o 9x®y = 7r and (ii) Qx®y (x 7T 0 0x ®y = ( 7 T 1 <8> 7T2 ) o ( 6 * 1 As 0y(y) = 0, 9x®y (x ® y)= 0. Case 2: x G ker 7Ti. There exists a finite subset A' of A such that x = A' wax a, where wa G R. It follows that 0 = 7Ti(x = I WaXa ) = ^2 VccE A' J a(zA' and di{x) = 61 f Waxa 1 = ^2 wa6l(xa) = Watl- \a£A' / a£A' a€A' neAcn£Aa where Aa and t^a are as defined in Lemma 2.2. As we have seen in the proof of Theorem 2.3, E « ga' Wat^a G ann M C annP, and hence Y2aeA' watnaP = 0 f°r all p G P. This implies 7r2 ( ^ aeA/ wat^ay) = 0, i.e. ^ aGA/ watm y G ker7r2. We infer from Lemma 1.5 that there exists an P-homomorphism 6 : F2 —* F2 such that (i) 7r2o0 = tt2 and (ii) 9 (J^aeA' w<*t»<*y) = 0, and hence X q6A' wat^aV G ker 0. Define : F -> F by 9x®y = 0x®y(z ® y) = 9x(x) <8> 9(y) = I ^ S w<*tw x v I ® ^ 2 nVl \a^A' fxeAa J 1=1 n = ^ 2 ^ 2 '5 2 w°‘tv<*ri(xn®yi). (*) « e A' fie Aa 1=1 Now, 0 = 0 £ Wat^ y I = Watna9(y) - ^ 2 (**) VqiGA' / agA' a£A' Z=1 PROJECTIVE, FLAT AND MULTIPLICATION MODULES 123 But F2 is a free i?-module. Thus for all 1 < / < n, the coefficient of yi in (**) is zero. Hence n Y. y ^ Wat pgr I = 0. aGA' 1=1 It follows from (*) that 0x®y (x 7T O 6x® y — (7Ti Following [17, p. 100], a submodule N of an P-module M is called a pure submodule if AT n aM = aN for all a G R. In particular, an ideal I of R is pure if Rx — Ix for every x G I. It is known that every pure ideal of a ring R is either locally zero or locally R. For, take M to be any maximal ideal of R. If I C M , then for all x G 7, ( R x ) m = I m (R x ) m Q Mm (R x ) m ? and hence (R x ) m = M m (R x ) m > and by Nakayama’s Lemma, ( R x ) m = 0m , and hence I m = 0m - On the other hand, if / ^ M , then I fl ( R\M ) ^ 0 , and hence I m — m R - It is well known that if I is a finitely generated multiplication ideal with pure annihilator, then I is a flat ideal [16, Theorem 2.2]. We generalize this result to multiplication modules (not necessarily finitely generated). We first give a lemma. Lemma 2.6. Let R be a ring and M a multiplication R-module such that ann M is a finitely generated pure ideal. Then M is a flat module. P roof. Let ann M = Y^i=\ R ai- Then for all i, there exists yi € ann M such that ai( 1 - yi) = 0. Put 1 - y = (1 - 2/i)(i-sfe) ---(l- yn)- Then y € ann M, and a^( 1 — y) = 0 . It follows that £ ann(l — y) for all i, and hence ann M C ann(l — y). On the other hand, as y M = 0, we infer that M = (1 — y )M and hence ann(l — y) Q ann M. This gives that ann M = ann(l — y) and that y is idempotent. Now, ann(l — y) is a pure ideal, and by [16, Theorem 2.2] i?(l — y) is a flat ideal of R. It follows by Theorem 2.5 that M Corollary 2.7. Let R be a ring and M a multiplication R-module. Suppose that ann M is a pure ideal. Then M is a flat module. P roof. Assume P is a maximal ideal of R. By [8 , Theorem 1.2] either M p = Op in which case it is obvious that M p is a flat Z?p- module, or there exist p € P and m G M such that (1 — p )M C Rm. It follows that (1 — p)Mann(m) = 0, and hence (1 — p)ann(m) C ann M. This implies that ann(m)p C (ann M )p C ann (M p) C ann(i?ra)p = ann(m)p, so that ann(Mp) = ann (M )p , see also [3, Theorem 2 .1 ]. As ann M is a pure ideal of R, we have that ann (M p) is a pure ideal of Rp. Since every pure ideal is idempotent and hence multiplication, we infer that ann (M p) is a principal ideal of 124 MAJID M. ALI AND DAVID J. SMITH Rp. By Lemma 2.6, M p is again a flat R p -module. Hence, M is locally flat and therefore M is flat [ 6 ]. □ We make three observations. First, as a finitely generated pure ideal is a prin cipal ideal generated by an idempotent, Corollary 2.4 shows that a multiplication .R-module M whose annihilator is a finitely generated pure ideal is projective. But every projective module is flat, [6], [7], [13]. Thus M is a flat module. This gives a different proof of Lemma 2.6. Second, from the proof of Corollary 2.7 it is obvi ous that a multiplication module whose annihilator is an idempotent ideal is a flat module. Third, Q is a flat Z-module, but is not multiplication, which shows that the converse of Corollary 2.7 is not true in general. Our final result of this chapter may be compared with [14, Theorems 2.1 and 2.2]. The proof is now straightforward. Corollary 2.8. Let R be a ring and M a multiplication R-module. If I is a projective (resp. flat) ideal of R such that annMPl/ = 0, then M ® I is a projective (resp. flat) R-module. 3. Sums of Multiplication Modules Let Z be the ring of integers and p a prime number. Consider the Priifer p-group Zpoo as a Z-module. Note that Zp can be identified with a submodule of Zp 2. In fact, pZ p2 = Zp. More generally, for all m : k > 1 with m > k, Zpk can be identified with a submodule of Z pm. Hence Zp C Zp2 C • • • C Zpoo and Zpoo = (J Zpfc. These Z pk are the only nonzero submodules of Zpoo, and each is fc>i a cyclic (hence multiplication) submodule. Also, for all m, k > 1, 7Lpk + Z pm = Zpr, where r = max{fc, m}, so that the sum of any two submodules of Zpoo is multipli cation. We note however that Zpoo is neither finitely generated nor multiplication. The following theorem gives several equivalent conditions for the sum of a collection of submodules in which the sum of any two distinct members is multiplication to be a multiplication module. It generalizes [18, Theorems 2 and 8 (i)]. Theorem 3.1. Let R be a ring and M an R-module. Let N\( A £ A) be a col lection of submodules of M such that N\ + N tl is multiplication for all \ ^ pi. Let N = and A = Y lx e A ^ x ■ N]. Then the following conditions ar-e equivalent: (i) A + ann(n) = R for all n G N. (ii) N\ = [Nx : N ]N for all A e A. (iii) For every finite sum H of two or more modules N\, H is multiplication and H = [H : N]N. (iv) N is multiplication. Proof, (i) = > (ii). Let A, ^ G A with A ^ H- By Lemma 1.1, N\ + = A(N\ + Nfj,). Let A G A and x G N\. Suppose that K = { r e R\rx G [Nx : A^]A^}. PROJECTIVE, FLAT AND MULTIPLICATION MODULES 125 Assume that K ^ R. Then there exists a maximal ideal P of R such that K C P. We discuss two cases. Case 1: AC. P. Then for all /i ^ A, N\ + — A(N\ + N^) C P(N\ + N^) C N\ + N^ so that N\ + ATM = P(N\ + N^). As N\ + iVM is multiplication, we infer that Rx = I(N\ + ) for some ideal I of R, and hence Rx = I(N\ + A g = IP(N\ + NJ = P(I(NX + iVM)) = P x. There exists p G P such that (1 — p)x = 0. It follows that (1 — p) G K C P, a contradiction. Case 2: A P. There exists fi G A such that [A^ : AT] P, and hence there exists q G P such that (1 — q)N C TV^. It follows that (1 — q)N\ Q N\ + N M and hence there exists an ideal J of R such that (1 — q)N\ = J(N\ + N ^). Next (1 - g) JN = J( 1 - g)AT C J(iVA + A g = (1 - g)iVA C Nx so that (1 — q)J C [N\ : AT], This implies that (1 - g f x € (1 - But this implies (1 — q)2 G K C P, a contradiction. Thus K — R and x G [AT\ : Ar]A',that is A ^ C [ATa : AT] AT. Since the other inclusion is always true, N\ = [N\ : AT]N. (ii) ==>• (iii). Let A' be a finite subset of A, and let H = J^agA' The fact that H is multiplication follows from [1, Theorem 2.3]. As N\ = [N\ : N ]N for all A G A', we have H = £ ^ = ( Z : ) N Q : N ^N- AgA' \AgA' / But [H : N ]N C H. Thus = [tf : AT]AT. (iii) ==>• (iv). Let i f be any submodule of M. For all x G K fl A”, there exists a finite subset Ax of A such that x G X]agax ^a = N x. It follows that KniV= X Pzc X KniV,. xGifrw xeKnN Nx is multiplication and N x = [ATX : AT] A/". Hence K n N C ^ [X : ATX]ATX C ^ [if : NX][NX : N]N C [X : AT] AT C i f f li V xGXnyv xG/cniV so that K D N = [K : N]N and N is multiplication. (iv) =>■ (i) follows by Lemma 1.1. This finishes the proof of the theorem. □ If R is a ring and N\(\ G A) a collection of submodules of an P-module M such that £ a g a -^a is multiplication, must all of N\ (or at least some of them) be multiplication modules? The answer is no, as the following example shows: Let R be a ring with distinct maximal ideals M\ and M 2. Then R — M\ + M 2 is a multiplication ideal but neither M\ nor M 2 is multiplication. (See for example [11, Example 38]). We show that the answer is yes under an additional condition. 126 MAJID M. ALI AND DAVID J. SMITH Theorem 3.2. Let R be a ring and N x(X G A) a collection of submodules of an R-module M such that N = X^agA multiplication. If, moreover, N x n is multiplication for all X, fi G A with A ^ /x, then all submodules N\(X G A) are multiplication modules. P roof. Let A = '^2Xe/i[Nx ■ N], By Lemma 1.1, N = A N . Let A 6 A. Let K C N\. Clearly [K : N X]NX C K. For the reverse inclusion, let y G K. Set H = { r e R\ry G [K : N X]NX}. Suppose that H ^ R. There exists a maximal ideal P of R such that H C P. Case 1: A C P. Then N = AN C PN C N so that N = PAT. Next, Ry = I N for some ideal I of R and hence Ry = I N = /FAT = P(IN) = Py. There exists p G F such that (1 — p)y = 0 and hence 1— p G F C P , a contradiction. Case 2: A <£. P. Then [JVa : N] + :N]£P. If [N\ : N]% P , there exists q G F such that (1 — q)N C N\. Now K = [K : TV] AT C [X : N X]N, and hence (1 - q)y 6 (1 - q )K c [K : JVA](1 - q)N C [K : ATa]ATa. It follows that 1 — q G H C P , also a contradiction. Finally if X ^ a ^ m ■ -^1 £ there exists /iG A and g' G P such that (1 — (1 - q')JNx = J( 1 - q')Nx C J (Nx fl N^) = (1 — 9')t f C K so that (1 — q')J C [A : A a ], and hence (1 - This gives a contradiction since (1 — q')2 G H C P. Therefore H = R and x G [K : Aa]Na so that K = [K : NX]NX. This shows that N x is a multipli cation module. □ It is well-known that if A and B are F-modules, then A ® B is projective if and only if A and B are projective, [10]. The same is not true for multiplication modules, for example, R is a multiplication F-module but R(B R is not. We apply Theorem 3.2 to give an alternative proof to [ 8 , Theorem 2.2] characterizing modules which are direct sums of multiplication modules. If N x(X G A) is a non-empty collection of F-modules and A G A, we let N x = ® Nn and N = ® N x. Note N = Nx ® Nx for all A G A. We iden- Ag A tify the N x with their natural injections in N. PROJECTIVE, FLAT AND MULTIPLICATION MODULES 127 Theorem 3.3. Let R be a ring and N x(X E A) a collection of R-modules. Then N = © N\ is multiplication if and only if: AeA (i) N\ is multiplication for all A G A. (ii) For each A G A, there exists an ideal A x of R such that A XNX = N x and A\N\ = 0 . Proof. Suppose first that N is multiplication. Then 0 = N\ f l N\ D N\ f l A^ D 0, so that N\ n = 0 which is a multiplication module for all A, fi G A, A ^ fi. From Theorem 3.2 it follows that N x is multiplication for all A G A. Let A G A. Then N\ = A \N = A\N\ + A xN xjo r some ideal A\ of R. Hence A\N\ Q NXC\ Nx = 0 so that N\ = A\N\ and A XNX = 0. Conversely, suppose the modules N x(X G A) satisfy (i) and (ii). Then Aa C ann(TVA) C [Nx : N x] = [Nx : N], and hence N x C A XN C [Nx : N}N C N x, so that N x = [Nx : N]N for all A G A. Put A = E X£A[NX : AT]. Then N = A N , and hence N x = [A^a : N ]N — [ATa : N ]A N = A N X for all A G A. But Nx is multiplication for all A G A. Thus, by Lemma 1.1, R = A + ann(x) for all x e Nx(X E A). Let n G N. There exist a finite subset A' of A and elements x x G N x(X G A') such that n = X^AeA' x *' ^ follows that R — A + P| ann(^A) = A + ann I R x a I C A + ann(n) AeA' \AeA' J so that R = A + ann(n). By Lemma 1.1, N is multiplication. This concludes the proof of the theorem. □ The next result should be compared with [ 8 , Corollary 2.3]. Corollary 3.4. Let R be a ring N x(X G A) a collection of finitely generated R-modules. Then N = © N x is multiplication if and only if AeA (i) N x is multiplication for all A G A. (ii) a n n (^ ) + a n n (^ ) = R for all A G A. Proof. Suppose that N is multiplication. Then (i) follows by Theorem 3.3. Let A G A. As N x D N x = 0, we infer from [1, Theorem 2.1] that R = ann (^Nx fl N x ^j = ann(A^A) + ann (j^xj + ann(a:) for all x G N and in particular for all x E N x. As N x is finitely generated, R = &nn(Nx) + a n n (^ ) + f] ann(x), and (ii) follows. Conversely, suppose the x £ N x modules N x(X E A) satisfy (i) and (ii). Let A G A. Then R C ann(^A) + [Nx : Nx\ = ann{Nx) + [Nx : iV] C ann(A^A) + : N] C R neA so that R = ann(Nx) + X^^eAt-^M • N] and by [18, Theorem 2 Corollary 2], N is multiplication. □ 128 MAJID M. ALI AND DAVID J. SMITH We close with the following property of the direct sum of a finite set of multipli cation modules. Corollary 3.5. Let R be a ring and N{(1 < i < n) a finite collection of R-modules. Then TV = ©TV; is multiplication if and only if: (i) TV, is multiplication for all i € {1,... , n}. (ii) ann(TV*) + ann(TV,) + ann(a) = R for all a G TV, + Nj and all i < j. Proof. The sufficiency is immediate by Theorem 3.3 and Corollary 3.4. Con versely, suppose the modules Ni satisfy (i) and (ii). Let i < j. Then [Ni : TV,-] + [Nj : TV,]+ann(a) = R and hence Ra = ([Ni : Nj] + [Nj : Ni])Ra for all a G TVj+TV,-. Summing over all a G Ni, we get Ni = [Ni :Nj]Ni + [Nj : TV,] TV,= [TV, : TV ,-]^ + TV,N n3 = [Ni : Nj]Ni + [Ni : TV,] AT, C[Ni : Nt + N3}(Ni + N3) C Nt, so that Ni = [Ni-.Ni + Nj^Ni + Nj). Similarly, Nj = [Nj : Ni + N j^N i + Nj). Let K be a submodule of Ni + N j. It follows from Lemma 1.1 that K n (Ni + Nj) = KDNi + KH Nj = [K : TV,]TV, [K+ : Nj]N3 = [K'.NiftNizNi + Nj^Ni + Nj) + [K : Nj][Nj : TV, + N3](Nt + TV,) C [K : TV, + TVj](TVi + TV,) C K fl (TV, + TV,), so that [i^ : TV, + TV,-](TV, + TV,) = K fl (TV, + TV,), and hence Ni + TV,- is multiplication. It follows from [1, Theorem 2.3] that TV is multiplication, and the proof is complete. □ In particular the case when A is finite in Corollary 3.4 (or the modules TVj are finitely generated in Corollary 3.5) is of interest. If TV* (1 < i < n) is a finite collection of finitely generated i?-modules, then TV = ©TV; is multiplication if and only if all Ni are multiplication and ann(TV^) + ann(TV,) = R for all i < j. In this case, ann(TV) is multiplication if and only if ann(TV^) are all multiplication, see [18, Theorem 8 (ii)] and [1, Theorem 2.3]. Also, using [ 1 , Corollary 2.4], it is easily verified that if ann(TV^) are finitely generated, then so too is ann(TV). References 1. M.M. Ali and D.J. Smith, Finite and infinite collections of multiplication mod ules, Beitrage zur Algebra und Geometrie, 42 (2001), 557-573. 2. D.D. Anderson, Some remarks on multiplication ideals II, Comm, in Algebra 28(2000), 2577-2583. 3. D.D. Anderson, Some remarks on multiplication ideals, Math. Japonica 4 (1980), 463-469. PROJECTIVE, FLAT AND MULTIPLICATION MODULES 129 4. D.D. Anderson, Multiplication ideals, multplication rings and the ring R (X ), Canad. J. Math. 28 (1976), 760-768. 5. A. Barnard, Multiplication modules , J. Algebra, 71 (1981), 174-178. 6 . N. Bourbaki, Commutative Algebra , Addison-Wesley, USA, 1972. 7. S.U. Chase, Direct products of modules, Trans AMS, 97 (1960), 457-473. 8. Z.A. El-Bast and P.F. Smith, Multiplication modules, Comm, in Algebra, 16 (1988), 755-779. 9. R. Gilmer, Multiplicative Ideal Theory , Kingston, Queen’s, 1992. 10. S.U. Hu, Introduction to Homological Algebra , Holden-Day, San Francisco, 1962. 1 1 . H.C. Hutchins, Examples of Commutative Rings , New Jersey, 1981. 12. I. Kaplansky, Commutative Rings , Boston, 1970. 13. M.D. Larsen and P.J. McCarthy, Multiplicative Theory of Ideals, New York, 1971. 14. A.G. Naoum, On the product of a module by an ideal , Beitrage zur Alg. und Geom. 27 (1988), 153-159. 15. A.G. Naoum, A. Mahmood, and F.H. Alwan, On projective and flat modules, Arab J. Math. 7 (1986), 41-53. 16. A.G. Naoum and M.A.K. Hasan, On finitely generated projective and flat ideals in commutative rings, Per. Math. Hungar, 16 (1985), 251-260. 17. P. Ribenboim, Algebraic Numbers, Wiley, 1972. 18. P.F. Smith, Some remarks on multiplication modules, Arch, der Math. 50(1988), 223-235. 19. W.W. Smith, Projective ideals of finite type , Canad. J. Math. 21 (1969), 1057-1061. Majid M. Ali David J. Smith Department of Mathematics Department of Mathematics Sultan Qaboos University University of Auckland Muscat Auckland OMAN NEW ZEALAND [email protected] [email protected] y) = 0, and this completes the proof of the theorem. □ y) — 0. We discuss two cases. Case 1: y G ker7r2. As P is flat, it follows by Lemma 1.4 that there exists an P-homomorphism 9y : F2 —»► F2 such that (i) 7t2 o 0y = 7r2 and (ii) 9y (y) = 0. Define 9x(g)y : F —> F by 0*®,, = 0i <8> 0y. Then 0. As 0(y) G F2, put 0(y) = ^ ”=1 riVi- Hence