Some Properties of S -Semiannihilator Small Submodules and S -Small Submodules with respect to a Submodule

Let R be a commutative ring with nonzero identity, S ⊆ R be a multiplicatively closed subset of R , and M be a unital R -module. In this article, we introduce the concepts of S -semiannihilator small submodules and S - T -small submodules as generalizations of S -small submodules. We investigate some basic properties of them and give some characterizations of such submodules, especially for (fnitely generated faithful) multiplication modules.


Introduction
Troughout this paper, R is a commutative ring with nonzero identity and M denotes a unital R-module.Also, S ⊆ R is a multiplicatively closed subset of R. We use the notations " ⊆ " and " ≤ " to denote inclusion and submodules, respectively.As usual, the rings of natural numbers, integers, and integer modulo n will be denoted by N, Z, and Z n , respectively.A module M over a ring R (not necessarily commutative) is called prime if for every nonzero submodule N of M, Ann(N) � Ann(M).An R-module M is called faithful if Ann(M) � 0. An R-module M is called a multiplication module, if for any submodule N of M, N � IM for some ideal I of R, and in this case, N � (N: M)M (see [1]).A submodule of an R-module M is called small (superfuous) which is denoted by N ≪ M, if for any submodule X of M, N + X � M, which implies that X � M. It is clear that the zero submodule of every nonzero module is small.More details about small submodules can be found in [2,3].In [4], the author introduced the concept of a semiannihilator small submodule of a module over a commutative ring R such that N is called semiannihilator small (sa-small for short), denoted by N ≪ sa M, if for every submodule X of M with N + X � M implies that Ann(X) ≪ R.An ideal I of R is an sa-small ideal of R if it is an sa-small submodule of R as an R-module.Let T be an arbitrary submodule of M. In [5], a submodule N is called a T-small submodule of M provided for each submodule X of M, T ⊆ N + X, which implies that T ⊆ X.
A nonempty subset S of R is called a multiplicatively closed subset of R if (i) 0 ∈ S, (ii) 1 ∈ S, and (iii) ss ′ ∈ S for all s, s ′ ∈ S, see [6].Let M be an R-module and S be a multiplicatively closed subset of R. Ten, M is called an S-multiplication module if for each submodule N of M, there exist s ∈ S and an ideal I of R such that sN ⊆ IM ⊆ N [7].Te concept of S-Noetherian rings has been introduced and investigated by Anderson et al. [8].Farshadifar introduced and studied in brief the notions of S-secondary submodules and S-copure submodules [9,10].S ¸engelen Sevim et al. in [11] described the concept of S-prime submodules.After, the generalizations of S-prime submodules have been studied in [12,13].Recently, the concept of S-small submodules have been studied in [14].Here, we introduce and study the notions of S-semiannihilator small submodules and S-T-small submodules as generalizations of T-small submodules.In Sections 2 and 3, various properties of such submodules are considered.

S-Semiannihilator Small Submodules
In this section, we defne the concept of S-semiannihilator small submodules of an R-module and we get some characterizations of them.
We begin with the following defnition.Defnition 1.Let M be an R-module.
(1) We say that a submodule N of M is an S-small submodule of M which is denoted by N ≪ S M if there exists s ∈ S such that whenever N + X � M for some submodule X of M, it implies that sM ⊆ X.We say that an R-module M is an S-hollow module if every submodule of M is an S-small submodule of M.
Remark 2. Te following results follow from the defnition: Moreover, in this case, M is an S-multiplication R-module because suppose that sM � 0 for some s ∈ S. It is sufcient to take I � (s); then, 0 � sN ⊆ IM ⊆ N for every submodule N of M, as needed.(2) Let M be an R-module and N < ⊕ M. Ten, there exists a proper submodule X of M such that M � N ⊕ X.If N ≪ S M, then there exists s ∈ S such that sM � sN + sX ⊆ X. Tis implies that sN � 0 and so S ∩ Ann(N) ≠ ∅. (3) It is clear that every small submodule is also S-small.
In particular, the zero submodule is an S-small submodule of M. Te following example shows that converse is not necessarily true in general.Clearly, if S ⊆ U(R) and N is an S-small submodule of M, then N is small.(4) If N ≪ S M, then for every submodule X of M with N + X � M, S ∩ (X: R N) ≠ ∅, since there exists s ∈ S such that sN ⊆ sM � sN + sX ⊆ X, as needed.
Example 1 (1) Consider M � Z ⊕ Z as a Z-module and take (2) Consider the Z-module M � Z 6 and the submodule N � 〈2〉.Take the multiplicatively closed subset Ten, N is an S-small submodule of M. Because we have N + 〈3〉 � Z 6 and N + Z 6 � Z 6 , let s � 3. Ten, sZ 6 ⊆ 〈3〉 and sZ 6 ⊆ Z 6 .But N is not a small submodule of M because N + 〈3〉 � Z 6 and 〈3〉 ≠ Z 6 .In general, let p, q be distinct prime numbers and consider the Z-module Z pq � Z p ⊕ Z q .Ten, the submodule N � 〈p〉 � 0 ⊕ Z q is an S-small submodule of M such that S � q n |n ∈  N ∪ 0 { }}.Moreover, the submodule Example 2. Consider M � Z p ⊕ Z p as a Z-module such that p is a prime number.It is clear that every proper submodule of M is prime, and for any submodule N of M, (N: Z M) � pZ.Also, M is a prime Z-module, and it is an Proposition 3. Let M be an R-module.Ten, the following statements are true: Proof.Te proofs are straightforward.
We recall that a submodule N of an R-module M is a semiannihilator small (briefy, sa-small) submodule if whenever N + X � M for some submodule X of M, implying that Ann(X) ≪ R.

□ Defnition 4
(1) A submodule N of M is called an S-semiannihilator small (briefy, S-sa-small) submodule of M which is denoted by N ≪ sa S M if there exists s ∈ S such that whenever N + X � M for some submodule X of M, it implies that sAnn(X) ≪ R.
(2) An ideal I of R is called S-semiannihilator small (briefy, S-sa-small) ideal if it is an S-semiannihilator small submodule of R as an R-module.

Lemma 5. Let M be an R-module and S, T be two multiplicatively closed subsets of R with
Proof.Te proof is straightforward.
Let S be a multiplicatively closed subset of R. Te saturation of S is the set S * � x ∈ R|xy ∈ S, for some y ∈ R  .It is clear that S * is a multiplicatively closed subset of R and that S ⊆ S * .□ Proposition 6.Let N be a submodule of an R-module M. Ten, N is an S-sa-small submodule of M if and only if N is an S * -sa-small submodule of M.
Proof.Let N be an S-sa-small submodule of M. Ten, by Lemma 5, N is an S * -sa-small submodule of M. Conversely, let N be an S * -sa-small submodule of M. Suppose N + X � M for some submodule X of M. Ten, there exists s * ∈ S * such that s * Ann(M) ≪ R. Ten, there exists r ∈ R such that 2 Journal of Mathematics Let M be an R-module and N, K be submodules of M. Ten, the following assertions hold: □ Proposition 8. Let M be an R-module and I be an ideal of R.
Ten, the following assertions hold: Hence, there exists s ∈ S such that sAnn(JM) ≪ R, since sAnn(J) ⊆ sAnn(JM), so sAnn(J) ≪ R. (ii) Let IM + X � M for some submodule X of M. We have X � JM for some ideal J of R. Tus, IM + JM � (I + J)M � M. By Nakayama's lemma, there exists a ∈ I + J such that (1 − a)M � 0. Since M is faithful, 1 − a � 0 and so a � 1. Tus, I + J � R.
Since I ≪ sa S R, there exists s ∈ S such that sAnn(J) � sAnn(JM) ≪ R, as needed.
S M. By the following example, we show that if f: M ⟶ M ′ is an epimorphism, then the image of an S-sa-small submodule of M need not be S-sa-small in M ′ .
Te following example shows that the sum of S-sa-small submodules of an R-module M need not be an S-sa-small submodule of M.
Example 5. Consider the Z-module Z and the multiplicatively closed subset S � Z − 3Z.Te submodules 2Z and 3Z are the S-sa-small submodules of Z.But 3Z + 2Z is not an S-sa-small submodule of Z.
□ Defnition 11.An R-module M is called an S-semiannihilator hollow (briefy, S-sa-hollow) module if every proper submodule of M is an S-sa-small submodule of M.
Example 6 (1) Consider the Z-module Z and the multiplicatively closed subset Ten, Z is an S-sahollow module, but it is not an S-hollow module.Because 3Z + 5Z � Z, but for any s ∈ S, sZ ⊈ 5Z.
Proposition 12. Let M, M ′ be R-modules and f: M ⟶ M ′ be an epimorphism.If M ′ is an S-sa-hollow module, then M is an S-sa-hollow module.
Proof.Let N be a submodule of M. Ten, f(N) is a submodule of M ′ .Since M ′ is an S-sa-hollow module, then f(N) ≪ sa S M ′ .Tus, f − 1 (f(N)) ≪ sa S M, and since

Journal of Mathematics
By assumption, there exists s ∈ S such that sAnn(F ⊗ K) ≪ R. Since F is a faithfully fat R-module, Ann(F ⊗ K) � Ann(K).Tus, sAnn(K) ≪ R, so N is an S-sa-small submodule of M.

S-Small Submodules with respect to a Submodule
Let T be an arbitrary submodule of an R-module M and S ⊆ R be a multiplicatively closed subset of R. In this section, we introduce and study another generalization of S-small and T-small submodules, namely, S-T-small submodules.

Defnition 15
(1) Let M be an R-module and T be an arbitrary submodule of M. A submodule N of M is called an S-T-small submodule of M which is denoted by N ≪ S− T M if there exists s ∈ S such that whenever T ⊆ N + X for some submodule X of M, it implies that sT ⊆ X.
We say that M is an S-T-hollow module if every submodule of M is S-T-small in M. (2) Let J be an ideal of R.An ideal I of R is called S-J-ideal of R if there exists s ∈ S such that whenever J ⊆ I + K for some ideal K of R, then sJ ⊆ K. R is an

S-T-hollow ring if it is an S-T-hollow as an R-module.
Observation 16.Let M be an R-module.
(1) Take T � M; then, N ≪ S− T M if and only if N ≪ S M.

) Clearly, every S-T-small submodule is a T-small
submodule, but the following example shows that converse is not necessarily true.(3) If T ⊆ N and N ≪ S− T M, then there exists s ∈ S such that s ∈ Ann(T), so S ∩ Ann(T) ≠ ∅.Equivalently, if for some submodule T of M, S ∩ Ann(T) � ∅, then either T ⊈ N or N ≪ S− T M.
Proposition 17.Let M be an R-module, L ≤ T ≤ M, and K ≤ M. Ten, the following assertions hold: We show that sM ⊆ X for some s ∈ S. We have , and hence, sT ⊆ X ∩ T for some s ∈ S because L ≪ S T. Tus, sT ⊆ X for some s ∈ S, so L ≪ S− T M.

□
Proposition 18.Let M be an R-module with submodules Proof.Let T ⊆ K + X for some submodule X of M. Tus, T ⊆ N + X, so there exists s ∈ S such that sT ⊆ X. Terefore, Proposition 21.Let M be an R-module and K and 0 ≠ T be submodules of M. Te following statements are equivalent: For every R-module N and R-homomorphism g: N ⟶ M, T ⊆ K + Im(g) implies that sT ⊆ Im (g) for some s ∈ S Proof.
(i) ⇒ (ii) and (ii) ⇒ (iii) are clear.(iii) ⇒ (i).Let T ⊆ K + X for some submodule X of M. Let i: X ⟶ M be the inclusion map.Ten, T ⊆ K + X ⊆ K + Im(i), and by (iii), there exists s ∈ S such that sT ⊆ Im(i) � X.

□
Proposition 22.Let M, M ′ be R-modules and f: M ⟶ M ′ be an R-homomorphism.If K and T are submodules of M such that K ≪ S− T M, then f(K) ≪ S− f(T) M ′ .In particular, if  (2)

□
Hence, there exists s ∈ S such that F ⊗ sT � s(F ⊗ T) ⊆ F ⊗ K. Tus, 0 ⟶ F ⊗ s T ⟶ F ⊗ K is exact, so 0 ⟶ sT ⟶ K is exact since F is faithfully fat.Tus, sT ⊆ K and N ≪ S− T M, as needed.

Conclusions
In this article, we introduced the concepts of S-semiannihilator small submodules and S-T-small submodules as generalizations of S-small submodules.We showed the concepts of annihilator small submodules and T-small submodules are diferent from the concept of S-small submodules.Several properties, examples, and characterizations of such submodules have been investigated.Moreover, we investigated the properties and the behavior of these structures under homomorphisms, Cartesian product, and localizations.

□ Example 7 .Corollary 13 .Theorem 1 .
(a)  We consider the Z and Z 4 as Z-modules, the multiplicatively closed subset S � Z − 0 { }, and the natural epimorphism π: Z ⟶ Z 4 .Ten, Z is an S-sa-hollow Z-module, and 0 { } ≪ sa S Z; π( 0 { }) � 0 is not an S-sa-small submodule of Z 4 .Because 0 + Z 4 � Z 4 , but for any s ∈ S, sAnn Z (Z 4 ) � s(4Z) ≪ Z.Let M be an R-module and N be a submodule of M. If M/N is an S-sa-hollow module, then M is an S-sahollow module.Proof.Apply Proposition 12.□ Let M be an R-module and N be a submodule of M. Assume that F is a faithfully fat R-module.Ten, if Terefore, K is an S − T-small submodule of M. □ Example 9. Consider the Z-homomorphism f: Z 10 ⟶ Z 20 with f(x) � 2x and the multiplicatively closed subset S � 3n : n ∈ N ∪ 0 { } { }.Ten, the submodule 〈10〉 is an S-small submodule of Z 20 , but f − 1 (〈10〉) � 〈5〉 is not S-small in Z 10 .Because 〈5〉 + 〈2〉 � Z 10 , sZ 10 ⊈ 〈2〉 for every s ∈ S.Theorem 23.Let M be a Noetherian R-module and N be a submodule of M. Ten, N is an S-T-small submodule of M if and only

Theorem 25 .
Let M be a fnitely generated faithful multiplication R-module and T ≤ M. Ten, N is an S-T-small submodule of M if and only if (N: R M) is an S-(T: M)-small ideal of R. Proof.Let N ≪ S− T M and (T: M) ⊆ (N: M) + J for some ideal J of R. Ten, (T: M)M ⊆ ((N: M) + J)M � (N: M)M + JM, and since M is a multiplication module, we have T ⊆ N + JM.Hence, there exists s ∈ S such that sT ⊆ JM, so s(T: M)M ⊆ JM.Since M is a cancellation module, s(T: M) ⊆ J. Terefore, (N: R M) is an S-(T: M)-small ideal of R. Conversely, let (N: R M) ≪ S− (T: M) R and T ⊆ N + X for some X ≤ M. Tus, (T: M)M ⊆ (N: M) M + (X: M)M � ((N: M) + (X: M)) M since M is a multiplication module, so (T: M) ⊆ (N: M) + (X: M) since M is a cancellation module.By assumption, there exists s ∈ S such that s(T: M) ⊆ (X: M), and hence, sT ⊆ X. Tis implies that N is an S-T-small submodule of M. □ Theorem 26.Let M be an R-module and N, T ≤ M. Assume that F is a faithfully fat R-module.If F ⊗ N is an S-F ⊗ T-small submodule of F ⊗ M, then N is an S-T-small submodule of M. Proof.Let F ⊗ N ≪ S− F⊗T F ⊗ M and T ⊆ N + K for some submodule K of M. Ten, F ⊗ T ⊆ F ⊗ (N + K) � F ⊗ N + F ⊗ K.