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Tensor Product of Faithful Multiplication Modules

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Tensor Product of Faithful Multiplication Modules International Journal of Algebra, Vol. 6, 2012, no. 4, 193 - 200 Tensor Product of Faithful Multiplication Modules Saeed Rajaee Department of Mathematics Payame Noor University of Mashhad, Iran saeed [email protected] Abstract All rings are commutative with identity and all modules are uni- tal. In this paper, we shall investigate some properties are conserved under tensor product operations. We introduce the prime submodules of tensor product of faithful multiplication modules on a commutative ring. Mathematics Subject Classification:13C05, 13C13, 13A15, 15A69 Keywords: multiplication module, cancellation module, faithful module, faithfully flat module, flat module, pure submodule, tensor product 1 Introduction Throughout this note all rings commutative with identity and M a unital R- module. Let A and B be two K-algebras and x be a nonzero-divisor element of A and y a nonzero-divisor element of B, then x⊗y is a nonzero-divisor element of A⊗K B.IfI be a proper ideal of A, then I ⊗K B is a proper ideal of A⊗K B. We recall that A⊗K B is a free (hence flat) extension of A and B.AnR-module M is called a multiplication module if for every submodule N of M there exists an ideal I of R such that N = IM, see [1], [2], [5], [11], [12]. An R-module M is called cancellation module if for ideals I and J of R, IM = JM implies that I = J and M is faithful if AnnR(M) = 0. We note that I ⊆ (N : M) and hence N = IM ⊆ (N : M)M ⊆ N,soN = IM =(N : M)M and if M be a cancellation module, then I =(N : M)=(IM : M). If M be a faithful multiplication module, then M is finitely generated ([3], Theorem 2.6]) and IM = M for any proper ideal I of R. A submodule N of R-module M is called prime submodule if N = M and for r ∈ R and m ∈ M, rm ∈ N implies that r ∈ (N :R M)orm ∈ N, see [4], [7], [9], [10]. An R-module M is said to be a prime module if Ann(N)=Ann(M) for each nonzero submodule N of M. 194 S. Rajaee M is called weak-cancellation module, whenever for ideals I and J of R, IM = JM implies that I + Ann(M)=J + Ann(M). Invertible ideals, finitely generated faithful multiplication modules and free modules are cancellation module. Let M be a multiplication R-module, then for every ideal I of R and submodule N of M,(N :M I)=(N :M IM). In particular, for N = 0, we have AnnM (I)=AnnM (IM) and if M be a faithful multiplication module (hence flat), then Ann(N)=Ann(N : M). Every projective (in particular, free) module is flat. If M is a module such that every finitely generated submodule of M is contained in a flat submodule (or itself be flat), then M is flat. If R be a regular (semisimple) ring, then every R-module M is flat. For finitely generated modules over a Noetherian local ring, flatness, projectivity, and freeness are all equivalent. An R-module M is flat if and only if for any finitely generated ideal I of R, the natural map I ⊗R M −→ IM is an isomorphism of abelian groups. Let 0 −→ N −→ M −→ K −→ 0 is an exact sequence in the category of R-modules, if N and K are flat modules, then M is too flat. Moreover, if M be flat, then K is flat if and only if N ∩ IM = IN for every (f.g.) ideal I of R, i.e. N is a pure submodule of M. Lazard and Govorov proved that flat modules are precisely the direct limits of (f.g.) free modules. The tensor product of projective (resp. flat, multiplication, faithful multi- plication, faithfully flat, finitely generated faithful multiplication) modules is a projective (resp. flat, multiplication, faithful multiplication, faithfully flat, finitely generated faithful multiplication) module but the converse is true un- der some conditions, see [5]. An R-module M is faithfully flat if M is flat and for all R-modules N, N ⊗M = 0 implies that N = 0. Cohn called a submodule N of M a pure submodule if the sequence 0 → N ⊗ E → M ⊗ E is exact for every R-module E, see [4]. Any direct summand N of an R-module M is pure, while the converse is true if M/N is of finite presentation [4]. Consequently, every subspace of a vector space over a field is pure, see [8]. Let M and M are faithful multiplication R-modules and N be a pure submodule of M and K a pure submodule of M , then N ⊗ K is pure submodule of M ⊗ M . A submodule N of M is multiplication if and only if K ∩ N =(K : N)N for all submodules K of M. An ideal I of R is called pure ideal if for every ideal J of R, IJ = I ∩ J. The pure submodules of flat modules are flat, from which it follows that pure ideals are flat ideals. If R be an integral domain and M a faithful multiplication R-module, then for every nonzero submodule N of M, Ann(N)=0. 2 Preliminary Notes Let M and M are faithful multiplication (hence f.g.) R-modules, then tensor product M ⊗R M is also a faithful multiplication R-module. Moreover if R Tensor product of faithful multiplication modules 195 be an integral domain, then every nonzero submodule N ⊗R K of M ⊗R M is faithful. A ring over which every module is projective is called semisimple. The k-fold tensor product are tensor powers M ⊗k as follows: ⊗0 ⊗1 ⊗2 ⊗3 M = R, M = M, M = M ⊗R M, M = M ⊗R M ⊗R M and so on. We note that if M is nonzero and finitely generated then M ⊗k =0 for all k. Theorem 2.1. [[10] Theorems 1.3, 1.5] Let N be a proper submodule of a multiplication R-module M. Then the following statements are equivalent: i) N is a prime submodule; ii) for every submodules K, L of M, we have KL ⊆ N =⇒ K ⊆ NorL⊆ N iii) for every m, n ∈ M,ifmn =(Rm)(Rn) ⊆ N then m ∈ N or n ∈ N. (iv) AnnR(M/N)=(N :R M)=P is a prime ideal of R. (v) N = PM for some prime ideal P of R with AnnR(M) ⊆ P . Theorem 2.2. [[3] Theorem 2.3, Corollary 2.4] Let M be a multiplication R-module. If N is a prime submodule of M, then there exists a unique prime ideal P of R with P ⊇ Ann(M) such that N = PM.IfP be a prime ideal of R, then PM ∈ Spec(M) if and only if (PM : M)=P . Theorem 2.3. [[5], Theorem 2] Let R be a ring and M, M be R-modules. i) If M and M are faithful multiplication then so is M ⊗ M . ii) If M and M are faithfully flat then so is M ⊗ M . iii) If M and M are finitely generated faithful multiplication then so is M⊗M . Definition 2.4. A submodule N of M is called multiplication, whenever for every submodule K ≤ M, K ∩ N =(K : N)N. By [5], if M and M are flat (in particular, faithful multiplication) R- modules, then for ideals I and J of R, ∼ ∼ (I ⊗ J)(M ⊗ M ) = IM ⊗ JM = JM ⊗ IM Theorem 2.5. Let R is a ring, M and M are faithful multiplication R- modules and N ≤ M and K ≤ M . i) If N and K are pure submodules of M and M respectively, then N ⊗ K is a pure submodule of M ⊗ M . ii) N ⊗ K is multiplication and idempotent submodule of M ⊗ M . iii) N ⊗ K is multiplication and L ⊗ T =((N ⊗ K):R (M ⊗ M ))(L ⊗ T ) for every submodule L ⊗ T of N ⊗ K. proof: Since M and M are faithful multiplication, hence by Theorem 2.3, M ⊗ M is a faithful multiplication R-module. Therefore Ann(M ⊗ M )=0 is pure, then by [6] Theorem 1.1, the statements can be obtained. 196 S. Rajaee 3 Main Results Let M is an R-module and M = N ⊕ K, then N and K are pure submodules of M, since for any ideal I of R, IM = IN ⊕ IK ⇒ N ∩ IM = N ∩ (IN ⊕ IK)=IN +(N ∩ IK)=IN Therefore N is a pure submodule of M. Similarly, K ≤ M is pure. Lemma 3.1. Let {Mλ}λ∈Λ be a family of R-modules, then i) if for every λ ∈ Λ, Mλ be a faithful multiplication R-module, then λ∈Λ Mλ is also a faithful multiplication R-module, ii) if for every λ ∈ Λ, Mλ be a faithfully flat R-module, then λ∈Λ Mλ is also a faithfully flat R-module, iii) if for everyλ ∈ Λ, Mλ be a finitely generated faithful multiplication R- module, then λ∈Λ Mλ is also finitely generated faithful multiplication R- module. proof: By Theorem 2.2, it is obvious. Corollary 3.2. Let M and M are finitely generated faithful multiplication R-modules and N ≤ M, K ≤ M , where N = IM and K = JM, then for every n ∈ N, ⊗n ⊗n ⊗n ⊗n ∼ ⊗n ⊗n ⊗n ⊗n ∼ ⊗n ⊗n (N K : M M ) = (N : M ) (K : M ) = I J in particular, if N or K be a flat submodule ⊗n ⊗n ⊗n ⊗n ∼ ⊗n ⊗n ⊗n ⊗n ∼ ⊗n ⊗n (N K : M M ) = (N : M )(K : M ) = I J ⊗n ⊗n ∼ ⊗n ⊗n ∼ K M = N M ⇔ K = N Theorem 3.3.
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