Tensor Product of Faithful Multiplication Modules

International Journal of Algebra, Vol. 6, 2012, no. 4, 193 - 200

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 unital. 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 multiplication, 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 under 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. Let M and M are ﬁnitely generated faithful multiplication R-modules. If N ≤ M and K ≤ M , with N = IM and K = JM, then S−1 N ⊗ K M ⊗ M ∼ S−1N ⊗ S−1K S−1M ⊗ S−1M ( :R ) = ( RS :RS RS ) Proof:Since M and M are ﬁnitely generated faithful modules, hence Ann S−1M Ann S−1M RS ( )= RS ( )=0

Similarly, since M and M are ﬁnitely generated multiplication R-modules, −1 −1 hence S M and S M are ﬁnitely generated multiplication RS-modules. Therefore −1 ∼ −1 S (N ⊗ K :R M ⊗ M ) = S (N : M) ⊗R (K : M ) Tensor product of faithful multiplication modules 197

∼ S−1 N M ⊗ S−1 K M ∼ S−1N S−1M ⊗ S−1K S−1M = ( :R ) RS ( :R ) = ( :RS ) RS ( :RS )

∼ S−1N ⊗ S−1K S−1M ⊗ S−1M ∼ S−1I ⊗ S−1J = ( RS :RS RS ) = RS Theorem 3.4. Let M and M are faithful multiplication (hence finitely generated) R-modules. If N be a prime submodule of M and K be a prime submodule of M , then there exists unique prime ideals P ⊇ Ann(M) and ∼ Q ⊇ Ann(M ) of R where N ⊗ K = PM⊗ QM . Moreover N ⊗ K is a prime submodule of M ⊗ M ,ifN or K be flat R-module and PQ∈ Spec(R). proof: Since M and M are finitely generated faithful multiplication R- modules and N ∈ Spec(M), K ∈ Spec(M ), there exist unique prime ideals P, Q ∈ Spec(R) such that N =(N : M)M = PM with P ⊇ Ann(M) and K =(K : M )M = QM with Q ⊇ Ann(M ), then ∼ P ⊗ Q ⊇ Ann(M) ⊗ Ann(M ) = Ann(M ⊗ M )

∼ N ⊗ K =(N : M)M ⊗ (K : M )M = [(N : M) ⊗ (K : M )](M ⊗ M )

∼ =(P ⊗ Q)(M ⊗ M ) = PM ⊗ QM We have also ∼ ∼ N ⊗ K = [(N : M) ⊗ (K : M )](M ⊗ M ) = (N ⊗ K : M ⊗ M )(M ⊗ M ) Since M ⊗M is a faithful nonzero R-module, hence PQ⊇ Ann(M ⊗M )=0. ∼ If K or N be ﬂat, then (N ⊗ K : M ⊗ M ) = (N : M)(K : M ), hence ∼ N ⊗ K = (PQ)(M ⊗ M ). By Theorem 2.1, if we have PQ ∈ Spec(R), then N ⊗ K ∈ Spec(M ⊗ M ). Corollary 3.5. Let M and M are (f.g.) faithful multiplication R-modules and P, Q be prime ideals of R, then ∼ i) (PM ⊗ M : M ⊗ M ) = (PM : M)=P , hence PM ⊗ M is a prime submodule of M ⊗ M . ∼ ∼ ii) (M ⊗ QM : M ⊗ M ) = (QM : M ) = Q, hence M ⊗ QM is a prime submodule of M ⊗ M . ∼ ∼ iii) for any proper ideal I of R, M ⊗M = I(M ⊗M ) = IM⊗M = M ⊗IM . iv) rad(M ⊗ M )=J(R)(M ⊗ M ). v) Let N ⊗ K be a submodule of M ⊗ M and I an ideal of R, then N ⊗ K is a multiplication R-module if and only if for any submodule L ⊗ T of M ⊗ M , (L ⊗ T : N ⊗ K)(N ⊗ K : M ⊗ M )=(L ⊗ T : M ⊗ M ) vi) for any ideal I of R, ∼ ∼ I =(I(M ⊗ M ):M ⊗ M ) = (IM ⊗ M : M ⊗ M ) = (M ⊗ IM : M ⊗ M ) Moreover N ⊗ K is a faithful submodule of M ⊗ M if and only if the ideal ∼ (N ⊗ K : M ⊗ M ) = (N : M) ⊗ (K : M ) be faithful. 198 S. Rajaee

Corollary 3.6. Let M and M are (f.g.) faithful multiplication free R- modules and N ≤ M, K ≤ M where N = IM and K = JM, then rad(N ⊗ K)= (N ⊗ K : M ⊗ M )(M ⊗ M )

√ ∼ = (N : M) ⊗ (K : M )(M ⊗ M )= I ⊗ J(M ⊗ M )

In particular, if N or K is ﬂat, then √ √ √ ∼ rad(N ⊗ K) = IJ(M ⊗ M )=( I ∩ J)(M ⊗ M )

Proof: We note that M ⊗M is a cancellation multiplication R-module, hence for any submodule N ⊗ K,

N ⊗ K =(N ⊗ K :R M ⊗ M )(M ⊗ M )

Since M and M are (f.g.) faithful multiplication R-modules, then

∼ (N ⊗ K : M ⊗ M ) = (N : M) ⊗ (K : M ) √ Now since M ⊗ M is free, hence rad(I(M ⊗ M )) = I(M ⊗ M ) for every ideal I of R.

Theorem 3.7. Let M and M are (f.g.) faithful multiplication free R- modules and N ≤ M, K ≤ M , where N = IM and K = JM, then ∼ i) rad(N ⊗ M ) = (radN) ⊗ M . ∼ ii) rad(M ⊗ K) = M ⊗ rad(K). Proof: i) Since M ⊗ M is a faithful multiplication free R-module, therefore √ ∼ rad(N ⊗ M )=rad(IM ⊗ M ) = rad(I(M ⊗ M )) = I(M ⊗ M )

√ ∼ = IM ⊗ M =(radN) ⊗ M ii) Similarly, we have √ ∼ rad(M ⊗ K)=rad(M ⊗ JM ) = rad(J(M ⊗ M )) = J(M ⊗ M )

√ ∼ = M ⊗ JM = M ⊗ rad(K)

In particular, if N or K be radical submodule of M or M respectively, then N ⊗ M or M ⊗ K is a radical submodule of M ⊗ M . Tensor product of faithful multiplication modules 199

Corollary 3.8. Let M and M are (f.g.) faithful multiplication free R- modules and N ≤ M, K ≤ M , where M = ⊕λ∈ΛMλ and N = ⊕λ∈ΛNλ, where Nλ is a submodule of Mλ for any λ ∈ Λ, then ∼ ∼ rad ( Nλ) ⊗ M = rad (Nλ ⊗ M ) = rad(Nλ ⊗ M ) λ∈Λ λ∈Λ λ∈Λ

∼ = (radNλ) ⊗ M λ∈Λ

ACKNOWLEDGEMENTS. This article was supported by the Grant of Payame Noor University of Iran. The author would like to thank the referee for his/her comments.

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Received: September, 2011