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Projective, Flat and Multiplication Modules

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Projective, Flat and Multiplication Modules 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 <ea PROJECTIVE, FLAT AND MULTIPLICATION MODULES 117 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.
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