Startseite Mathematik On weakly classical 1-absorbing prime submodules
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On weakly classical 1-absorbing prime submodules

  • Zeynep Yılmaz ORCID logo EMAIL logo , Bayram Ali Ersoy , Ünsal Tekir , Suat Koç ORCID logo und Serkan Onar ORCID logo
Veröffentlicht/Copyright: 24. November 2025
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Open Mathematics
Aus der Zeitschrift Open Mathematics Band 23 Heft 1

Abstract

In this paper, we study weakly classical 1-absorbing prime submodules of a nonzero unital module M over a commutative ring R having a nonzero identity. A proper submodule N of M is said to be a weakly classical 1-absorbing prime submodule, if for each m ∈ M and nonunits a, b, c ∈ R, 0 ≠ abcm ∈ N implies that abm ∈ N or cm ∈ N. We give various examples and properties of weakly classical 1-absorbing prime submodules. Also, we investiage the weakly classical 1-absorbing prime submodules of tensor product F ⊗ M of a (faithfully) flat R-module F and any R-module M. Also, we prove that if every proper submodule of an R-module M is weakly classical 1-absorbing prime, then Jac(R)3 M = 0. In terms of this result, we characterize modules over local rings in which every proper submodule is weakly classical 1-absorbing prime.

Keywords: weakly classical prime submodules; weakly classical 1-absorbing prime submodules; weakly classical 2-absorbing submodules
MSC 2020: 13A15; 13C05; 13C13

1 Introduction

In this article, all rings under consideration are assumed to be commutative with nonzero identity and all modules are assumed to be nonzero unital. Let R always denote such a ring and M denote such an R-module. A proper submodule N of M is called a prime submodule if whenever xm ∈ N for some x ∈ R and m ∈ M, then x ∈ (N: R M) or m ∈ N, where (N: R M) is the annihilator Ann(M/N) of R-module M/N, that is, (N: R M) = {x ∈ R:xM ⊆ N}. In this case, (N: R M) is a prime ideal of R. However, the converse is not true in general. For instance, consider Z -module Z ⊕ Q and note that N = 2 Z × Z is not a prime submodule since 2 0 , 1 2 ∈ N , 2 ∉ N : Z Z ⊕ Q and 0 , 1 2 ∉ N . But N : Z Z ⊕ Q = 0 is a prime ideal of Z . Several authors have extended the notion of prime ideals to modules (see, [1], [2], [3], [4]). In 2004, Behboodi and Koohy introduced the concept of weakly prime submodules (see, [5], 6]). According to [6], a proper submodule N of M is called a weakly prime submodule if whenever xym ∈ N for some x, y ∈ R and m ∈ M, then xm ∈ N or ym ∈ N. By definition, it is clear that every prime submodule is also weakly prime but the converse is not true in general. For instance, the submodule N = 2 Z ⊕ 0 of Z -module Z ⊕ Q is a weakly prime submodule which is not prime. It can be easily seen that a proper submodule N of M is weakly prime if and only if (N: R m) = {x ∈ R:xm ∈ N} is a prime ideal of R for every m ∉ N. So far, the concept of weakly prime submodules has drawn attention of many authors and has been studied in many papers. For example, Azizi in his paper [7], Lemma 3.2 and Theorem 3.3] studied the weakly prime submodules of tensor product F ⊗ M for a flat (faithfully flat) R-module F and any R-module M. Here, we should mention that after some initial works, many authors studied the same notion under the name “classical prime submodule”, and in this paper we prefer to use classical prime submodule instead of weakly prime submodule. In 2016, Mostafanasab et al. gave a new generalization of classical prime submodules which is called weakly classical prime submodule as follows: a proper submodule N of M is called a weakly classical prime submodule if whenever 0 ≠ abm ∈ N for some a, b ∈ R and m ∈ M, then am ∈ N or bm ∈ N [8]. It is clear that zero submodule in any module is trivially weakly classical prime. However, zero submodule need not be classical prime in general. For instance, take the zero submodule N = ( 0 ̄ ) of Z -module Z 9 (where Z 9 denotes the factor ring Z / 9 Z ). Then N is weakly classical prime which is not classical prime. In a recent study, Yassine et al. gave a new generalization of prime ideal which is called 1-absorbing prime ideal and they characterized local rings in terms of 1-absorbing prime ideals. A proper ideal Q of R is said to be a 1-absorbing prime ideal if whenever xyz ∈ Q for some nonunits x, y, z ∈ R, then xy ∈ Q or z ∈ Q [9]. For more on 1-absorbing prime ideals, see [10], [11], [12]. Afterwards, Koç et al. defined the concept of weakly 1-absorbing prime ideals as follows: a proper ideal Q of R is said to be a weakly 1-absorbing prime if whenever 0 ≠ xyz ∈ Q for some nonunits x, y, z ∈ R, then xy ∈ Q or z ∈ Q [13]. Among many results in the paper [13], the authors studied weakly 1-absorbing prime ideals in the ring of real valued continuous functions C(X) on a topological space X, and also characterized rings in which every proper ideal is weakly 1-absorbing prime. According to [13], Theorem 14], every proper ideal of a ring R is a weakly 1-absorbing prime if and only if (R, M) is a local ring with M 3 = 0 or R is a direct product of two fields. Motivated by the studies of [8], 13], in this paper we introduce and study weakly classical 1-absorbing prime submodules of modules over commutative rings. A proper submodule N of M is said to be a weakly classical 1-absorbing prime submodule if whenever 0 ≠ xyzm ∈ N for some nonunits x, y, z ∈ R and m ∈ M, then xym ∈ N or zm ∈ N.

In the following table, we provide a compilation of the definitions of various families of submodules (and related ideals) that are directly related to the discussions and results developed in this work. These definitions constitute the fundamental building blocks employed throughout the paper, and their proper understanding is crucial for a clear comprehension of the subsequent analyses and proofs. Here, N is considered as a proper submodule of the R-module M, and Q be a proper ideal of the ring R.

Classical prime submodule abm ∈ N ⇒ am ∈ N or bm ∈ N.
Weakly classical prime submodule 0 ≠ abm ∈ N ⇒ am ∈ N or bm ∈ N.
1-absorbing prime ideal xyz ∈ Q with x, y, z nonunits ⇒ xy ∈ Q or z ∈ Q.
Weakly 1-absorbing prime ideal 0 ≠ xyz ∈ Q with x, y, z nonunits ⇒ xy ∈ Q or z ∈ Q.
Weakly classical 1-absorbing prime submodule 0 ≠ xyzm ∈ N with x, y, z nonunits ⇒ xym ∈ N or zm ∈ N.

For the sake of completeness, now we will give some notions and notations which will be followed in the sequel. Let R be a ring and M be an R-module. For every element x ∈ R (m ∈ M), principal ideal (cyclic submodule) generated by x ∈ R (m ∈ M) is denoted by Rx (Rm). Also, for any subset K of M and any submodule N of M, the residual of N by K will be designated by (N: R K) = {x ∈ R:xK ⊆ N}. In particular, if N = (0) is the zero submodule and K = {m} is a singleton for some m ∈ M, we prefer to use Ann R (m) instead of ((0): R K). Also, we prefer to use Ann R (M) to denote the annihilator of M instead of ((0): R M) [14]. Recall from [14] that M is said to be a faithful module if Ann R (M) is the zero ideal of R. Recall from [15] that an R-module M is said to be a multiplication module if every submodule N of M has the form IM for some ideal I of R. In this case, N = (N: R M)M. Ameri in his paper [16] defined product of two submodules of multiplication modules as follows: Let M be a multiplication module and N = IM, K = JM be two submodules of M. Then the product of N and K is defined by NK = IJM. This product is independent of the represetantions of N and K (see, [16], Theorem 3.4]). In particular, we use Nm to denote the product of N and Rm for every submodule N of M and m ∈ M. Also for any submodule N of M and any subset J of R, the residual of N by J is defined by (N: M J) = {m ∈ M: Jm ⊆ N}.

Among other results in this paper, we investigate the relations between weakly classical 1-absorbing prime submodules and weakly 1-absorbing prime ideals (see, Theorems 1, 2 and Corollary 1). We investigate the stability of weakly classical 1-absorbing prime submodules under homomorphism, in localization and in factor modules (see, Theorems 3, 4 and Corollaries 2, 3). Recall from that [17] that a proper submodule N of M is said to be a weakly 1-absorbing prime submodule if whenever 0 ≠ abm ∈ N for some nonunits a, b ∈ R and m ∈ M, then ab ∈ (N: R M) or m ∈ N. Recall from [18] that a proper submodule N of M is said to be a weakly semiprime if for each a ∈ R and m ∈ M, 0 ≠ a 2 m ∈ N implies that am ∈ N. An R-module M is said to be a reduced module if whenever a 2 m = 0 for some a ∈ R and m ∈ M, then am = 0 [19]. In Proposition 1, we investigate the relations between weakly classical 1-absorbing prime submodules and other classical submodules such as weakly 1-absorbing prime submodules, weakly classical prime submodules and weakly semiprime submodules. Also, we prove that in any cyclic module, weakly 1-absorbing prime submodules and weakly classical 1-absorbing prime submodules are equal (see, Proposition 2). We give two very long lists whose all items are characterizing weakly classical 1-absorbing prime submodules (see, Theorems 6 and 9). Also, we determine weakly classical 1-absorbing prime submodules of multiplication modules (Theorem 10). In algebra, Prime avoidance lemma states that if an ideal is contained in a finite union of prime ideals, then it must be contained in one of those primes. If we replace prime ideals with any ideals, then the claim is not true in general. For instance, consider R = Z 2 [ x , y , z ] / ( x , y , z ) 2 and I = { 0 ̄ , x ̄ , y ̄ , x + y ̄ } . Let P = { 0 ̄ , x ̄ } , Q = { 0 ̄ , y ̄ } and J = { 0 ̄ , x + y ̄ } . Then I ⊆ P ∪ Q ∪ J and I is not contained in one of the ideals P, Q, J of R. One can naturally ask for which rings, a similar version of prime avoidance lemma holds for any ideals: “When does I ⊆ ⋃ k = 1 n J k imply I ⊆ J k for some 1 ≤ k ≤ n ”. In 1975, Quartararo and Butts investigated the algebraic properties of such rings. According to [20] a ring R is said to be a u-ring if an ideal is contained in a finite union of ideals, then it must be contained in one of those ideals. Also, R is called a um-ring if M is an R-module and M = ⋃ i = 1 n N i for some submodules N i of M, then M = N i for some 1 ≤ i ≤ n. The authors in [20] showed that Prüfer domains and Bėzout rings are examples of u-rings. We give a characterization of weakly classical 1-absorbing prime submodules of modules over um-rings (see, Theorem 9). Furthermore, we investigate the weakly classical 1-absorbing prime submodules of direct product of modules (see, Theorems 11 and 12). Recall from [21] that an R-module F is said to be a flat R-module if for each exact sequence K → L → M of R-modules, the sequence F ⊗ K → F ⊗ L → F ⊗ M is also exact. Also, F is said to be a faithfully flat, the sequence K → L → M is exact if and only if F ⊗ K → F ⊗ L → F ⊗ M is exact. Azizi in [7], Lemma 3.2] showed that if M is an R-module, N is a submodule of M and F is a flat R-module, then (F ⊗ N: F⊗M a) = F ⊗ (N: M a) for every a ∈ R. In Theorem 13, we characterize weakly classical 1-absorbing prime submodules of tensor product F ⊗ M of a (faithfully) flat R-module F and any R-module M. Recall from [14] that the intersection of all maximal ideals of a ring R is denoted by Jac(R) which is called the Jacobson radical of R. Finally, we show that for any R-module M, if every proper submodule of M is weakly classical 1-absorbing prime submodule, then Jac(R)3 M = 0. In terms of this result, for any module M over a local ring ( R , m ) , every proper submodule of M is weakly classical 1-absorbing prime if and only if m 3 M = 0 (see, Theorem 14).

2 Characterization of weakly classical 1-absorbing prime submodules

Throghout this section, M will denote a nonzero unital module over a commutative ring R having a nonzero identity.

Definition 1.

A proper submodule N of an R-module M is called a weakly classical 1-absorbing prime if whenever 0 ≠ abcm ∈ N for some nonunits a, b, c ∈ R and m ∈ M, then abm ∈ N or cm ∈ N.

Recall from [22] that a proper submodule N of M is said to be a classical 1-absorbing prime submodule if abcm ∈ N for some nonunits a, b, c ∈ R and m ∈ M implies that abm ∈ N or cm ∈ N. By definition, it is clear that every classical 1-absorbing submodule is also weakly classical 1-absorbing submodule. However, the converse is not true in general. See the following example.

Example 1.

(Weakly classical 1-absorbing prime ⇏ Classical 1-absorbing prime). Let p, q, r be three distinct prime numbers and consider Z -module M = Z pqr . Suppose that N = ( 0 ̄ ) . Then N is clearly a weakly classical 1-absorbing. Since p ⋅ q ⋅ r ⋅ 1 ̄ ∈ N , p ⋅ q ⋅ 1 ̄ ∉ N and r ⋅ 1 ̄ ∉ N , it follows that N is not classical 1-absorbing prime submodule.

The following example shows that there exist nonzero weakly classical 1-absorbing prime submodule which is not classical 1-absorbing prime.

Example 2.

Consider Z 6 -module M = ∏ i = 1 ∞ Z 6 and the submodule N = ⨁ i = 1 ∞ Z 6 . Clearly N consists of all sequence x = ( x n ) n ∈ N where x n ≠ 0 ̄ for finitely many n ∈ N . Choose x = ( x n ) n ∈ N ∈ M − N . Since all ideals of Z 6 are ( 0 ̄ ) , ( 2 ̄ ) , ( 3 ̄ ) and Z 6 , we get ( N : R x ) = ( 0 ̄ ) , ( 2 ̄ )  or  ( 3 ̄ ) which are weakly 1-absorbing prime ideals. Then by Theorem 1, N is clearly a weakly classical 1-absorbing prime submodule of M. Now, take x = ( x n ) n ∈ N as constant sequence 1 ̄ , that is, x n = 1 ̄ for all n ∈ N . Then clearly ( N : R x ) = ( 0 ̄ ) which is not a 1-absorbing prime ideal. By [22], Theorem 2 (ix)], N is not a classical 1-absorbing prime submodule of M.

Theorem 1.

Let M be an R-module and N a proper submodule of M. If (N: R m) is a weakly 1-absorbing prime ideal of R for each m ∈ M\N, then N is a weakly classical 1-absorbing prime submodule of M. The converse is true provided that Ann R (m) = 0.

Proof.

Let (N: R m) be a weakly 1-absorbing prime ideal of R for each m ∈ M\N, and choose 0 ≠ abcm ∈ N for some nonunits a, b, c ∈ R and m ∈ M. If m ∈ N, then we are done. So, suppose that m ∈ M\N. Hence 0 ≠ abc ∈ (N: R m) and by assumption, we have ab ∈ (N: R m) or c ∈ (N: R m), that is, abm ∈ N or cm ∈ N. For the converse, suppose that N is a weakly classical 1-absorbing prime submodule of M, and choose m ∈ M\N such that Ann R (m) = 0. Let 0 ≠ abc ∈ (N: R m) for some nonunits a, b, c ∈ R. Then we get 0 ≠ abcm ∈ N. Since N is a weakly classical 1-absorbing prime submodule of M, we have either abm ∈ N or cm ∈ N. Hence ab ∈ (N: R m) or c ∈ (N: R m). Consequently, (N:m) is a weakly 1-absorbing prime ideal of R. □

In the converse of previous theorem, the condition “Ann R (m) = 0” is necessary. On the other words, even if N is a weakly classical 1-absorbing prime submodule of M and m ∈ M\N with Ann R (m) ≠ 0, then (N: R m) may not be weakly 1-absorbing prime ideal of R. See, the following example.

Example 3.

Let p, q be two distinct prime numbers and consider Z -module M = Z p 2 q 2 . Let N = ( 0 ̄ ) and m = p ̄ . Then note that N is a weakly classical 1-absorbing prime submodule and m ∈ M\N with Ann Z ( m ) ≠ 0 . However, ( N : Z M ) = p q 2 Z is not a weakly 1-absorbing prime ideal of Z .

Let M be an R-module. Recall from [14] that the set of all torsion elements is denoted by T R (M) = {m ∈ M:Ann R (m) ≠ 0}. It is worthwhile to note that T R (M) is not always a submodule of M. For instance, if we consider R = Z × Z -module M = Q × Q , then clearly T R ( M ) = 0 × Q ∪ Q × 0 is not a submodule of M. If T R (M) = M, M is said to be a torsion module. If there exists m ∈ M − T R (M), then we say that M is a non-torsion module.

Theorem 2.

Let M be a non-torsion module and N a proper submodule of M such that T R (M) ⊆ N. Then N is a weakly classical 1-absorbing submodule of M if and only if (N: R m) if a weakly 1-absorbing prime ideal of R for each m ∈ M\N.

Proof.

Assume that T R (M) ⊆ N. Then note that M − N ⊆ M − T R (M) which means that Ann R (m) = 0 for each m ∈ M − N. The rest follows from Theorem 1. □

Corollary 1.

Let R be a ring and I a proper ideal of R. Then I is a weakly classical 1-absorbing prime submodule of R-module R if and only if I is a weakly 1-absorbing prime ideal of R.

Proof.

Let I be a weakly classical 1-absorbing prime submodule of R-module R. Then by Theorem 1, we have (I: R 1) = I is a weakly 1-absorbing prime ideal of R. Now, let I be a weakly 1-absorbing prime ideal and choose nonunits a, b, c ∈ R and m ∈ M such that 0 ≠ abcm = ab(cm) ∈ I. Since I is a weakly 1-absorbing prime ideal, we have ab ∈ I or cm ∈ I. Which implies that abm ∈ I or cm ∈ I. □

Theorem 3.

Let M, M′ be two R-modules and f:M → M′ be an R-homomorphism.

(1) Suppose that f is a monomorphism and N′ is a weakly classical 1-absorbing prime submodule of M′ with f − 1 N ′ ≠ M , then f − 1 N ′ is a weakly classical 1-absorbing submodule of M.

(2) Suppose that f is an epimorphism and N is a weakly classical 1-absorbing prime submodule of M containing Ker(f), then f N is a weakly classical 1-absorbing prime submodule of M′.

Proof.

(1): Suppose that N′ is a weakly classical 1-absorbing prime submodule of M′ with f − 1 N ′ ≠ M . Let 0 ≠ a b c m ∈ f − 1 N ′ for some nonunits a, b, c ∈ R and m ∈ M. Since f is a monomorphism, 0 ≠ f(abcm) = abcf(m) ∈ N′. Since N′ is a weakly classical 1-absorbing prime submodule, we have abf(m) ∈ N′ or cf(m) ∈ N′. Which implies that f(abm) ∈ N′ or f(cm) ∈ N′. Therefore, a b m ∈ f − 1 N ′ or c m ∈ f − 1 N ′ . Consequently, f − 1 N ′ is a weakly classical 1-absorbing prime submodule of M.

(2): Assume that N is a weakly classical 1-absorbing prime submodule of M. Let a, b, c ∈ R be nonunits and m′ ∈ M′ be such that 0 ≠ a b c m ′ ∈ f N . By assumption, there exists m ∈ M such that m′ = f(m) and so 0 ≠ abcm′ = f(abcm) ∈ f(N). Since Ker(f) ⊆ N, we have 0 ≠ abcm ∈ N. As N is a weakly classical 1-absorbing prime submodule of M, we have either abm ∈ N or cm ∈ N. Hence, f(abm) = abm′ ∈ f(N) or f(cm) = cm′ ∈ f(N). Consequently, f(N) is a weakly classical 1-absorbing prime submodule of M′. □

As an immediate consequence of Theorem 3 2 , we have the following explicit result.

Corollary 2.

Let M be an R-module and L ⊆ N be two submodules of M. If N is a weakly classical 1-absorbing prime submodule of M, then N/L is a weakly classical 1-absorbing prime submodule of M/L.

The converse of previous corollary need not be true. See, the following example.

Example 4.

Let p be a prime number and consider Z -module M = Z p 3 3 . Let L = Z p 3 × ( 0 ̄ ) × ( 0 ̄ ) = N . Then N/L is a zero submodule which is clearly weakly classical 1-absorbing prime. Since p 3 ( 1 ̄ , 1 ̄ , 1 ̄ ) ∈ N and p 2 ( 1 ̄ , 1 ̄ , 1 ̄ ) ∉ N , it follows that N is not a weakly classical 1-absorbing prime submodule of M.

Corollary 3.

Let K and N be two submodules of M with K ⊆ N ⊆ M. If K is a weakly classical 1-absorbing prime submodule of M and N/K is a weakly classical 1-absorbing prime submodule of M/K, then N is a weakly classical 1-absorbing prime submodule of M.

Proof.

Assume that K is a weakly classical 1-absorbing prime submodule of M and N/K is a weakly classical 1-absorbing prime submodule of M/K. Let a, b, c ∈ R be nonunits and m ∈ M with 0 ≠ abcm ∈ N. Assume that abcm ∈ K. Since K is a weakly classical 1-absorbing prime submodule, we get either abm ∈ K ⊆ N or cm ∈ K ⊆ N. Now, assume that abcm ∉ K. Then we obtain 0 ≠ abc(m + K) ∈ N/K. Since N/K is a weakly classical 1-absorbing prime submodule, we have ab(m + K) ∈ N/K or c(m + K) ∈ N/K which implies that abm ∈ N or cm ∈ N, as needed. □

Theorem 4.

Let M be an R-module, N be a submodule of M and S be a multiplicative subset of R. If N is a weakly classical 1-absorbing prime submodule of M such that S −1 N ≠ S −1 M, then S −1 N is a weakly classical 1-absorbing prime submodule of S −1 M.

Proof.

Let N be a weakly classical 1-absorbing prime submodule of M and S −1 N ≠ S −1 M. Suppose that 0 ≠ a s b t c u m v ∈ S − 1 N for some nonunits a s , b t , c u ∈ S − 1 R and m v ∈ S − 1 M . Then there exists w ∈ S such that 0 ≠ wabcm ∈ N. Since N is a weakly classical 1-absorbing prime, then we have ab(wm) ∈ N or c(wm) ∈ N. Thus a s b t m v = w a b m w s t v ∈ S − 1 N or c u m v = w c m w u v ∈ S − 1 N . Therefore, S −1 N is a weakly classical 1-absorbing prime submodule of S −1 M. □

Proposition 1.

Let N be a proper submodule of M.

(1) If N is a weakly 1-absorbing prime submodule of M, then N is a weakly classical 1-absorbing prime submodule of M.

(2) If N is a weakly classical prime submodule of M, then N is a weakly classical 1-absorbing prime submodule and weakly semiprime submodule of M.

(3) Suppose that M is a reduced module. Then N is a weakly classical prime submodule of M if and only if N is a weakly semiprime submodule and weakly classical 1-absorbing prime submodule of M.

Proof.

(1), (2): Clear.

( 3 ) ⇒ : follows from (2). ⇐ : Suppose that N is a weakly classical 1-absorbing prime submodule and weakly semiprime submodule of a reduced module M. Choose a, b ∈ R and m ∈ M such that 0 ≠ abm ∈ N. Without loss of generality, we may assume that a, b are nonunits. Since M is reduced, we have 0 ≠ a 2 bm ∈ N. As N is a weakly classical 1-absorbing prime submodule, we get either a 2 m ∈ N or bm ∈ N. If a 2 m ∈ N, then am ∈ N because N is weakly semiprime. Thus, N is weakly classical prime submodule of M. □

Proposition 2.

Let M be a cyclic R-module. Then a proper submodule N of M is a weakly 1-absorbing prime submodule if and only if it is weakly classical 1-absorbing prime submodule of M.

Proof.

⇒ Follows from Proposition 1 (1). ⇐ Let M = Rm for some m ∈ M and N be a weakly classical 1-absorbing prime submodule of M. Suppose that 0 ≠ abx ∈ N for some nonunits a, b ∈ R and x ∈ M. If Rx = Rm = M, then we conclude that ab ∈ (N: R M). So assume that Rx ≠ M. Then there exists nonunit c ∈ R such that x = cm. Therefore 0 ≠ abx = abcm ∈ N. Since N is a weakly classical 1-absorbing prime submodule of M, we have either abm ∈ N or cm ∈ N. Hence ab ∈ (N: R M) or x = cm ∈ N. Consequently, N is a weakly 1-absorbing prime submodule of M. □

The following definition shows that whenever N is a weakly classical 1-absorbing prime submodule of M which is not a classical 1-absorbing prime submodule, there exist some nonunits a, b, c ∈ R and m ∈ M such that N has an (a, b, c, m) classical 1-quadruple-zero.

Definition 2.

Let N be a proper submodule of M and a, b, c be nonunits in R, m ∈ M. If N is a weakly classical 1-absorbing prime submodule and abcm = 0, abm ∉ N, cm ∉ N, then a , b , c , m is called a classical 1-quadruple-zero of N.

Theorem 5.

Let N be a weakly classical 1-absorbing prime submodule of M, and suppose that abcK ⊆ N for some nonunits a, b, c ∈ R and some submodule K of M. If a , b , c , k is not a classical 1-quadruple-zero of N for every k ∈ K, then abK ⊆ N or cK ⊆ N.

Proof.

Suppose that a , b , c , k is not a classical 1-quadruple-zero of N for every k ∈ K and abcK ⊆ N. Assume on the contrary that abK ⊈ N and cK ⊈ N. Then there are k 1, k 2 ∈ K such that abk 1 ∉ N and ck 2 ∉ N. If abck 1 ≠ 0, then we have ck 1 ∈ N, because abk 1 ∉ N and N is a weakly classical 1-absorbing prime. So assume that abck 1 = 0. Since a , b , c , k 1 is not a classical 1-quadruple-zero of N, we conclude that ck 1 ∈ N. In both cases, we have ck 1 ∈ N. Likewise, abk 2 ∈ N. On the other hand, note that abc(k 1 + k 2) ∈ N. A similar argument above shows that ab(k 1 + k 2) ∈ N or c(k 1 + k 2) ∈ N. As abk 2 ∈ N and ck 1 ∈ N, we have abk 1 ∈ N or ck 2 ∈ N which both of them are contradictions. Hence, we obtain abK ⊆ N or cK ⊆ N. □

Definition 3.

Let N be a weakly classical 1-absorbing prime submodule of M, and suppose that IJLK ⊆ N for some proper ideals I, J, L of R and some submodule K of M. We say that N is a free classical 1-quadruple-zero with respect to IJLK if a , b , c , k is not a classical 1-quadruple-zero of N for every elements a ∈ I, b ∈ J, c ∈ L and k ∈ K.

Remark 1.

Let N be a weakly classical 1-absorbing prime submodule of M and suppose that IJLK ⊆ N for some proper ideals I, J, L of R and some submodule K of M such that N is a free classical 1-quadruple-zero with respect to IJLK. Hence, if a ∈ I, b ∈ J, c ∈ L and k ∈ K, then abk ∈ N or ck ∈ N.

Corollary 4.

Let N be a weakly classical 1-absorbing prime submodule of M, and suppose that IJLK ⊆ N for some proper ideals I, J, L of R and some submodule K of M. If N is a free classical 1-quadruple-zero with respect to IJLK, then IJK ⊆ N or LK ⊆ N.

Proof.

Suppose that N is a free classical 1-quadruple-zero with respect to IJLK. Assume that IJK ⊈ N and LK ⊈ N. Then there are a ∈ I, b ∈ J, c ∈ L with abK ⊈ N and cK ⊈ N. Since abcK ⊆ N and N is a free classical 1-quadruple-zero with respect to IJLK, then Theorem 5 implies that abK ⊆ N or cK ⊆ N, which is a contradiction. Consequently, we have IJK ⊆ N or LK ⊆ N. □

Let M be an R-module and N be a submodule M. For every a ∈ R, m ∈ M : a m ∈ N is denoted by N : M a . It is easy to see that N : M a is a submodule of M containing N. In the next theorem, we give a long list whose each item characterizes weakly classical 1-absorbing prime submodules.

Theorem 6.

Let M be an R-module and N be a proper submodule M. The following conditions are equivalent.

(1) N is a weakly classical 1-absorbing prime submodule;

(2) For every nonunits a, b, c ∈ R, N : M a b c = 0 : M a b c ∪ N : M a b ∪ N : M c ;

(3) For every nonunits a, b ∈ R and m ∈ M with abm ∉ N; N : R a b m = 0 : R a b m ∪ N : R m ;

(4) For every nonunits a, b ∈ R and m ∈ M with abm ∉ N; N : R a b m = 0 : R a b m or N : R a b m = N : R m ;

(5) For every nonunits a, b ∈ R, m ∈ M and every proper ideal I of R with 0 ≠ abIm ⊆ N, either abm ∈ N or Im ⊆ N;

(6) For every nonunits a, b ∈ R, m ∈ M and every proper ideal I of R with 0 ≠ abIm ⊆ N, either aIm ⊆ N or bm ∈ N;

(7) For every proper ideal I of R, nonunit a ∈ R and m ∈ M with a I m ⊈ N, N : R a I m = 0 : R a I m or N : R a I m = N : R m ;

(8) For every proper ideals I, J, K of R and m ∈ M with 0 ≠ IJKm ⊆ N, either IJm ⊆ N or Km ⊆ N.

Proof.

(1) ⇒ (2): Suppose that N is a weakly classical 1-absorbing prime submodule of M. Let m ∈ N : M a b c . Then abcm ∈ N. If abcm = 0, then m ∈ 0 : M a b c . Assume that abcm ≠ 0. Since N is a weakly classical 1-absorbing prime submodule, we have abm ∈ N or cm ∈ N. Hence m ∈ N : M a b or m ∈ N : M c and so m ∈ 0 : M a b c ∪ N : M a b ∪ N : M c . Consequently, N : M a b c = 0 : M a b c ∪ N : M a b ∪ N : M c .

(2) ⇒ (3): Let abm ∉ N for some nonunits a, b ∈ R and m ∈ M. Assume that c ∈ N : R a b m . Then abcm ∈ N, and so m ∈ N : M a b c . Thus by part (2), m ∈ 0 : M a b c ∪ N : M a b ∪ N : M c . Since abm ∉ N, then m ∉ N : M a b which implies that m ∈ 0 : M a b c or m ∈ N : M c . Therefore c ∈ 0 : R a b m or c ∈ N : R m . Consequently, c ∈ 0 : R a b m ∪ N : R m . Thus N : R a b m = 0 : R a b m ∪ N : R m .

(3) ⇒ (4): By the fact that if an ideal (a subgroup) is the union of two ideals (two subgroups), then it is equal to one of them.

(4) ⇒ (5): Assume that 0 ≠ abIm ⊆ N for some nonunits a, b ∈ R, m ∈ M and a proper ideal I of R. Hence I ⊆ (N: R abm) and I ⊈ (0: R abm). If abm ∈ N, then we are done. So, assume that abm ∉ N. Therefore by part (4), we have that I ⊆ N : R m which implies that Im ⊆ N.

(5) ⇒ (6): Let 0 ≠ abIm ⊆ N for some proper ideal I of R, nonunits a, b ∈ R and m ∈ M. Assume that aIm ⊈ N. Then there exists x ∈ I such that axm ∉ N. Then note that ax(Rb)m ⊆ N. If 0 ≠ ax(Rb)m, then by part (5) we have (Rb)m ⊆ N which completes the proof. So assume that ax(Rb)m = 0. Since 0 ≠ abIm, there exists y ∈ I such that ay(Rb)m ≠ 0. This implies that 0 ≠ a(x + y)(Rb)m ⊆ N. As 0 ≠ ay(Rb)m ⊆ N, again by part (5), we have aym ∈ N or (Rb)m ⊆ N. Let aym ∈ N. Then we have a(x + y)m ∉ N. As 0 ≠ a(x + y)(Rb)m ⊆ N, by part (5), we get (Rb)m ⊆ N which implies that bm ∈ N. In the other case, we have bm ∈ N which completes the proof.

(6) ⇒ (7): Let a I m ⊈ N for some nonunit a ∈ R, some proper ideal I of R and m ∈ M. Choose b ∈ (N: R aIm). Then we have ab I m ⊆ N. If ab I m = 0, then we get b ∈ (0: R aIm). Now, assume that ab I m ≠ 0. Then by part (6), we get bm ∈ N which implies that b ∈ (N: R m). Thus, we conclude that N : R a I m = 0 : R a I m ∪ ( N : R m ) . This implies that N : R a I m = 0 : R a I m or N : R a I m = N : R m .

(7) ⇒ (8): Let 0 ≠ IJKm ⊆ N for some proper ideals I, J, K of R and m ∈ M. Assume that IJm ⊈ N. Then there exists a ∈ J such that a I m ⊈ N. Since aIKm ⊆ N, we have K ⊆ N : R a I m = 0 : R a I m or K ⊆ N : R a I m = N : R m by part (7). This gives aIKm = 0 or Km ⊆ N. In the second case, we are done. So we may assume that aIKm = 0. Since 0 ≠ IJKm, there exists b ∈ J such that IbKm ≠ 0. Since K ⊆ N : R b I m , again by part (7), we have bIm ⊆ N or Km ⊆ N. So assume that bIm ⊆ N. Then (a + b)Im ⊈ N and 0 ≠ (a + b)IKm ⊆ N. Since K ⊆ (N: R (a + b)Im), by part (7), we conclude that Km ⊆ N.

(8) ⇒ (1): Clear. □

Theorem 7.

Let N be a weakly classical 1-absorbing prime submodule of M and suppose that a , b , c , m is a classical 1-quadruple-zero of N for some nonunits a, b, c ∈ R and m ∈ M. Then

(1) abcN = 0 = ab(N: R M)m.

(2) If a, b ∉ (N: R cm), then a c ( N : R M ) m = b c ( N : R M ) m = a ( N : R M ) 2 m = b ( N : R M ) 2 m = c ( N : R M ) 2 m = 0 . In particular, ( N : R M ) 3 m = 0 .

Proof.

(1): Let abcN ≠ 0. Then there exists n ∈ N with abcn ≠ 0. This gives 0 ≠ abcn = abc(m + n) ∈ N. Since N is weakly classical 1-absorbing prime, we have ab(m + n) ∈ N or c(m + n) ∈ N. Which implies that abm ∈ N or cm ∈ N, which is a contradiction. Thus, abcN = 0. Now, we will show that ab(N: R M)m = 0. Choose x ∈ (N: R M) and assume that abxm ≠ 0. Then we have 0 ≠ abxm = ab(c + x)m ∈ N. If c + x is unit, we conclude that abm ∈ N which is a contradiction. Thus c + x is nonunit. As N is a weakly classical 1-absorbing prime, we have abm ∈ N or (c + x)m ∈ N. Then we obtain abm ∈ N or cm ∈ N, again a contradiction. Thus, ab(N: R M)m = 0. (2): Let a, b ∉ (N: R cm), that is, acm ∉ N and bcm ∉ N. Choose x ∈ (N: R M) and acxm ≠ 0. Then we have 0 ≠ a(b + x)cm ∈ N. Note that b + x must be nonunit. As N is weakly classical 1-absorbing prime, a(b + x)m ∈ N or cm ∈ N which yields that abm ∈ N or cm ∈ N. This is a contradiction. Hence, we have ac(N: R M)m = 0. Likewise, we have bc(N: R M)m = 0. Now, we will show that a ( N : R M ) 2 m = 0 . Suppose that a ( N : R M ) 2 m ≠ 0 . Then there exist x, y ∈ (N: R M) such that axym ≠ 0. Then by ab(N: R M)m = ac(N: R M)m = 0, we have 0 ≠ axym = a(b + x)(c + y)m ∈ N. Since abm ∉ N and acm ∉ N, (b + x) and (c + y) are nonunits. As N is weakly classical 1-absorbing prime, we have a(b + x)m ∈ N or (c + y)m ∈ N. Which implies that abm ∈ N or cm ∈ N. This is a contradiction. Thus, we have a ( N : R M ) 2 m = 0 . Similarly one can prove that b ( N : R M ) 2 m = c ( N : R M ) 2 m = 0 . Now, we will show that ( N : R M ) 3 m = 0 . Suppose to the contrary. Then there exist x, y, z ∈ (N: R M) such that xyzm ≠ 0. Then we have 0 ≠ (a + x)(b + y)(c + z)m = xyzm ∈ N. Also, (a + x), (b + y), (c + z) are nonunits. Then we conclude that (a + x)(b + y)m ∈ N or (c + z)m ∈ N. This implies that abm ∈ N or cm ∈ N which is a contradiction. Hence, ( N : R M ) 3 m = 0 . □

A submodule N of an R-module M is called a nilpotent submodule if ( N : R M ) k N = 0 for some positive integer k [23]. In particular, an element m ∈ M is nilpotent in M if Rm is a nilpotent submodule of M.

Theorem 8.

If N is a weakly classical 1-absorbing prime submodule of M that is not classical 1-absorbing prime submodule, then there exists classical 1-quadruple-zero a , b , c , m of N. If a, b∉(N: R cm), then ( N : R M ) 3 N = 0 and thus N is nilpotent. Furthermore, if M is a multiplication module, then N 4 = 0.

Proof.

Suppose that N is a weakly classical 1-absorbing prime submodule of M that is not a classical 1-absorbing prime submodule. Then there exists a classical 1-quadruple-zero a , b , c , m of N for some nonunits a, b, c ∈ R and m ∈ M. Suppose that a, b ∉ (N: R cm). Now, we will show that ( N : R M ) 3 N = 0 . Suppose that ( N : R M ) 3 N ≠ 0 . Then there are x, y, z ∈ (N: R M) and n ∈ N such that xyzn ≠ 0. By Theorem 7, we have 0 ≠ (a + x)(b + y)(c + z)(m + n) = xyzn ∈ N. Since N is a weakly classical 1-absorbing prime submodule, we have (a + x)(b + y)(m + n) ∈ N or (c + z)(m + n) ∈ N. Then we have abm ∈ N or cm ∈ N which is a contradiction. Thus, we obtain ( N : R M ) 3 N = 0 and N is nilpotent. Now, suppose that M is a multiplication module. Then it is clear that N 4 = (N: M)3 N = 0. □

Proposition 3.

Let M be a multiplication module and N a proper submodule of M. The following conditions are equivalent.

(1) N is a weakly classical 1-absorbing prime submodule of M.

(2) If 0 ≠ KLPm ⊆ N for some proper submodules K, L, P of M and m ∈ M, then either KLm ⊆ N or Pm ⊆ N.

Proof.

(1) ⇒ (2): Let 0 ≠ KLPm ⊆ N for some proper submodules K, L, P of M and m ∈ M. Since M is a multiplication module, there are ideals I, J, Q of R such that K = IM, L = JM and P = QM. Since 0 ≠ KLPm ⊆ N we have 0 ≠ IJQm ⊆ N, and thus we have IJm ⊆ N or Qm ⊆ N. Hence KLm ⊆ N or Pm ⊆ N.

(2) ⇒ (1): Suppose that 0 ≠ IJQm ⊆ N for some proper ideals I, J, Q of R and some m ∈ M. It is sufficient to set K := IM, L := JM and P := QM in part (2). Hence 0 ≠ KLPm ⊆ N implies that KLm ⊆ N or Pm ⊆ N. This implies that IJm ⊆ N or Qm ⊆ N. Consequently, N is a weakly classical 1-absorbing prime submodule of M by Theorem 6. □

Now, we are ready to give a new characterization of weakly classical 1-absorbing prime submodules of modules over um-rings.

Theorem 9.

Let R be a um-ring and M be an R-module. For any proper submodule N of M, the followings are equivalent.

(1) N is a weakly classical 1-absorbing prime submodule.

(2) For every nonunits a, b, c ∈ R, (N: M abc) = (N: M ab) or (N: M abc) = (0: M abc) or (N: M abc) = (N: M c).

(3) For every nonunits a, b, c ∈ R and every submodule L of M, 0 ≠ abcL ⊆ N implies that abL ⊆ N or cL ⊆ N.

(4) For every nonunits a, b ∈ R and every submodule L of M with abL ⊈ N, (N: R abL) = (0: R abL) or (N: R abL) = (N: R L).

(5) For every nonunits a, b ∈ R, every proper ideal J of R and every submodule L of M with 0 ≠ abJL ⊆ N implies that abL ⊆ N or JL ⊆ N.

(6) For every nonunits a ∈ R, every proper ideals I, J of R and every submodule L of M with 0 ≠ aIJL ⊆ N implies that aIL ⊆ N or JL ⊆ N.

(7) For every proper ideals I, J of R and every submodule L of M with IJL ⊈ N, (N: R IJL) = (0: R IJL) or (N: R IJL) = (N: R L).

(8) For every proper ideals I, J, K of R and every submodule L of M, 0 ≠ IJKL ⊆ N implies that IJL ⊆ N or KL ⊆ N.

Proof.

(1) ⇒ (2): Let m ∈ (N: M abc). Then we have abcm ∈ N. If abcm = 0, then m ∈ (0: M abc). Now, assume that abcm ≠ 0. As N is a weakly classical 1-absorbing prime submodule, we have abm ∈ N or cm ∈ N. Thus, we have m ∈ (N: M ab) ∪ (N: M c). This gives (N: M abc) = (N: M ab) ∪ (N: M c) ∪ (0: M abc). As R is a um-ring, we have (N: M abc) = (N: M ab) or (N: M abc) = (0: M abc) or (N: M abc) = (N: M c).

(2) ⇒ (3): Let 0 ≠ abcL ⊆ N. Then we have L ⊆ (N: M abc). Since 0 ≠ abcL, (N: M abc) ≠ (0: M abc). This gives L ⊆ (N: M abc) = (N: M ab) or L ⊆ (N: M abc) = (N: M c). Then we have abL ⊆ N or cL ⊆ N.

(3) ⇒ (4): Suppose that abL ⊈ N and take x ∈ (N: R abL). Then we have abxL ⊆ N. If abxL = 0, then we get x ∈ (0: R abL). Now, assume that abxL ≠ 0. Then by part (3), we conclude that xL ⊆ N, that is, x ∈ (N: R L). This gives that (N: R abL) = (0: R abL) ∪ (N: R L). The follows from the fact that if a submodule is a union of two submodules, then it must be equal to one of them.

(4) ⇒ (5): Let 0 ≠ abJL ⊆ N and assume that abL ⊈ N. Then we have J ⊆ (N: R abL) and (N: R abL) ≠ (0: R abL). Thus by part (4), we get J ⊆ (N: R abL) = (N: R L). This implies that JL ⊆ N which proves (5).

(5) ⇒ (6): Suppose that 0 ≠ aIJL ⊆ N and JL ⊈ N. Now, take x ∈ I. Since 0 ≠ aIJL, there exists y ∈ I such that ayJL ≠ 0. As 0 ≠ ayJL ⊆ N, by part (5), ayL ⊆ N. If 0 ≠ axJL ⊆ N, then similarly we have axL ⊆ N. If 0 = axJL, then 0 ≠ ayJL = a(x + y)JL ⊆ N. Then similar argument shows that a(x + y)L ⊆ N which implies that axL ⊆ N. Hence, we conclude that aIL ⊆ N.

(6) ⇒ (7): Suppose that IJL ⊈ N and a ∈ (N: R IJL). Then aIJL ⊆ N and there exists x ∈ J such that xIL ⊈ N. Assume that aIJL = 0. Then we have a ∈ (0: R IJL). Now, assume that aIJL ≠ 0. Then there exists y ∈ J such that 0 ≠ ayIL = yI(Ra)L ⊆ N. Then by part (6), we conclude that yIL ⊆ N or (Ra)L ⊆ N. In the second case, we have a ∈ (N: R L). So assume that yIL ⊆ N. If 0 ≠ axIL = xI(Ra)L ⊆ N, then by part (6), we have (Ra)L ⊆ N which implies that a ∈ (N: R L). So we may assume that axIL = 0. In this case, we have 0 ≠ ayIL = a(x + y)IL = (x + y)I(Ra)L ⊆ N. Again by part (6), we conclude that (x + y)IL ⊆ N or (Ra)L ⊆ N. If (x + y)IL ⊆ N, then we get xIL ⊆ N because yIL ⊆ N. This is a contradiction. Thus we conclude that (Ra)L ⊆ N, that is, a ∈ (N: R L). By above arguments, we conclude that (N: R IJL) ⊆ (0: R IJL) ∪ (N: R L). Since the other inclusion is always true, we have the equality (N: R IJL) = (0: R IJL) ∪ (N: R L). In this case, we have (N: R IJL) = (0: R IJL) or (N: R IJL) = (N: R L).

(7) ⇒ (8): Suppose that 0 ≠ IJKL ⊆ N and IJL ⊈ N. Then we have K ⊆ (N: R IJL) ≠ (0: R IJL). Then by part (7), we have K ⊆ (N: R IJL) = (N: R L) which implies that KL ⊆ N.

(8) ⇒ (1): It is clear. □

By [22], Theorem 4], we know that if N is a classical 1-absorbing prime submodule of M, then (N: R M) is a 1-absorbing prime ideal of R. One can naturally ask whether this property can be extended to weakly classical 1-absorbing prime or not. Now, we give a negative answer to this question with the following example.

Example 5.

Let p be a prime number and consider Z -module Z p 2 . Assume that N = ( 0 ̄ ) . Then N is trivially weakly classical 1-absorbing prime submodule. However, N : Z Z p 2 = p 2 Z is not a weakly 1-absorbing prime ideal.

The following is an immediate consequences of Theorem 6.

Corollary 5.

Let N be a proper submodule of M. Then N is a weakly classical 1-absorbing prime submodule of M if and only if for every nonunits a, b ∈ R and m ∈ M, (N: R abm) = (0: R abm) or (N: R abm) = (N: R m) or (N: R abm) = R.

Theorem 10.

Let M be a finitely generated multiplication module such that Ann R (M) is a 1-absorbing prime ideal of R. For a proper submodule N of M, the followings are equivalent.

(1) N is a weakly classical 1-absorbing prime submodule of M.

(2) If 0 ≠ N 1 N 2 N 3 N 4 ⊆ N for some proper submodules N 1, N 2, N 3 of M and some submodule N 4 of M, then N 1 N 2 N 4 ⊆ N or N 3 N 4 ⊆ N.

(3) (N: R M) is a weakly 1-absorbing prime ideal of R.

(4) N = PM for some weakly 1-absorbing prime ideal P of R with Ann R (M) ⊆ P.

Proof.

(1) ⇒ (2): Suppose that N is a weakly classical 1-absorbing prime submodule of M. Let 0 ≠ N 1 N 2 N 3 N 4 ⊆ N for some proper submodules N 1, N 2, N 3 of M and some submodule N 4 of M. Since M is multiplication, there exist proper ideals I 1, I 2, I 3 of R such that N i = I i M for every i = 1, 2, 3. Then we have 0 ≠ I 1 I 2 I 3 N 4 ⊆ N. Then by Theorem 9, I 1 I 2 N 4 = N 1 N 2 N 4 ⊆ N or I 3 N 4 = N 3 N 4 ⊆ N.

(2) ⇒ (1): It is similar to (1) ⇒ (2).

(1) ⇒ (3): Suppose that N is a weakly classical 1-absorbing prime submodule, a, b, c ∈ R are nonunits and 0 ≠ abc ∈ (N: R M). Then we have abcM ⊆ N. If abcM = 0, then ab ∈ Ann R (M) or c ∈ Ann R (M). Which implies that abM ⊆ N or cM ⊆ N. Now, assume that 0 ≠ abcM ⊆ N. Then by Theorem 9, abM ⊆ N or cM ⊆ N which implies that ab ∈ (N: R M) or c ∈ (N: R M). Hence, (N: R M) is a weakly 1-absorbing prime ideal.

(3) ⇒ (4): It is clear.

(4) ⇒ (1): Let 0 ≠ IJKL ⊆ N = PM for some proper ideals I, J, K of R and some submodule L of M. Then we have IJK(L: R M) ⊆ (PM: R M) = P + Ann R (M) = P by [24], Corollary to Theorem 9]. As P is a weakly 1-absorbing prime ideal, we have IJ(L: R M) ⊆ P or K(L: R M) ⊆ P. Then we have IJL ⊆ N or KL ⊆ N. Then by Theorem 9, N is a weakly classical 1-absorbing prime submodule of M. □

Since the proof of the following lemma is similar to the previous proposition (1) ⇒ (3), we omit the proof.

Lemma 1.

Let M be a faithful R-module and N a weakly classical 1-absorbing prime submodule of M. Then (N: R M) is a weakly 1-absorbing prime ideal of R.

Theorem 11.

Let M 1, M 2 be two R-modules and N 1 be a proper submodule of M 1. The following conditions are equivalent.

(1) N = N 1 × M 2 is a weakly classical 1-absorbing prime submodule of M = M 1 × M 2.

(2) N 1 is a weakly classical 1-absorbing prime submodule of M 1 and if whenever (a, b, c, m) is a classical 1-quadruple-zero of N 1 for some nonunits a, b, c ∈ R and m ∈ M 1, then abc ∈ Ann R (M 2).

Proof.

(1) ⇒ (2): Suppose that N = N 1 × M 2 is a weakly classical 1-absorbing prime submodule of M = M 1 × M 2. Let 0 ≠ abcm ∈ N 1 for some nonunits a, b, c ∈ R and m ∈ M 1. Then we have (0, 0) ≠ abc(m, 0) = (abcm, 0) ∈ N. Since N is a weakly classical 1-absorbing prime submodule of M, we have ab(m, 0) = (abm, 0) ∈ N or c(m, 0) = (cm, 0) ∈ N. This implies that abm ∈ N 1 or cm ∈ N 1. Thus, N 1 is a weakly classical 1-absorbing prime submodule of M 1. Now, assume that (a, b, c, m) is a classical 1-quadruple-zero of N 1. Then we have abcm = 0, abm ∉ N 1 and cm ∉ N 1. Now, we will show that abc ∈ Ann R (M 2). Suppose that abc ∉ Ann R (M 2). Then there exists n ∈ M 2 such that abcn ≠ 0. This gives (0, 0) ≠ abc(m, n) = (0, abcn) ∈ N. As N is a weakly classical 1-absorbing prime, we have ab(m, n) ∈ N or c(m, n) ∈ N. Which implies that abm ∈ N 1 or cm ∈ N 1. Both of them are contradictions. Thus, abc ∈ Ann R (M 2).

(2) ⇒ (1): Let a, b, c ∈ R be nonunits and m , n ∈ M = M 1 × M 2 be such that 0,0 ≠ a b c m , n ∈ N = N 1 × M 2 . First assume that abcm ≠ 0. Then by part (2), abm ∈ N 1 or cm ∈ N 1. So, we conclude that a b m , n ∈ N or c m , n ∈ N . This shows that N is a weakly classical 1-absorbing prime. So assume that abcm = 0. If abm ∉ N 1 and cm ∉ N 1, then (a, b, c, m) is a classical 1-quadruple-zero of N 1. Then by part (2), abcn = 0 and so abc(m, n) = (0, 0) which is a contradiction. Thus, we have abm ∈ N 1 or cm ∈ N 1. This argument shows ab(m, n) ∈ N or c(m, n) ∈ N which completes the proof. □

Proposition 4.

Let M 1, M 2 be two R-modules and N 1, N 2 be two proper submodules of M 1 and M 2, respectively. If N = N 1 × N 2 is a weakly classical 1-absorbing prime submodule of M = M 1 × M 2, then N 1 is a weakly classical 1-absorbing prime submodule of M 1 and N 2 is a weakly classical 1-absorbing prime submodule of M 2.

Proof.

One can easily verify the claim by a similar argument in the previous theorem. □

The following example shows that the converse of previous proposition is not true in general.

Example 6.

Let R = Z , M = Z 2 and N = p Z × q Z where p, q are two distinct prime integers. Since, p Z , q Z are prime ideals of Z , then p Z , q Z are weakly classical 1-absorbing prime Z -submodules of Z . Notice that 0,0 ≠ p p q 1,1 = p p q , p p q ∈ N , but neither p p 1,1 ∈ N nor q 1,1 ∈ N . So, N is not a weakly classical 1-absorbing prime submodule of M.

Let R i be a commutative ring with identity and M i be an R i -module for every i = 1, 2. Let R = R 1 × R 2. Then M = M 1 × M 2 is an R-module with componentwise addition and scalar multiplication. Also each submodule of M is in the form of N = N 1 × N 2 for some submodules N 1 of M 1 and N 2 of M 2.

Theorem 12.

Let M i be a faithful multiplication R i -module such that Ann R (M i ) is not unique maximal ideal of R i for each i = 1, 2. Suppose that R = R 1 × R 2 is a decomposable ring and M = M 1 × M 2. Further, assume that N = N 1 × N 2 ≠ 0 for some submodule N i of M i . The following statements are equivalent.

(1) N is a weakly classical 1-absorbing prime submodule of M.

(2) N = N 1 × M 2 for some classical prime submodule N 1 of M 1 or N = M 1 × N 2 for some classical prime submodule N 2 of M 2.

(3) N is a classical prime submodule of M.

(4) N is a weakly classical prime submodule of M.

Proof.

(1) ⇒ (2): Suppose that N = N 1 × N 2 is a weakly classical 1-absorbing prime submodule of M. Since M 1, M 2 are faithful modules, so is M. As N is a weakly classical 1-absorbing prime submodule, by Lemma 1, (N: R M) is a weakly 1-absorbing prime ideal of R. Since N is nonzero and M is faithful multiplication module, ( N : R M ) = ( N 1 : R 1 M 1 ) × ( N 2 : R 2 M 2 ) is nonzero. Then by the proof of [13], Theorem 10], we have ( N 1 : R 1 M 1 ) = R 1 or ( N 2 : R 2 M 2 ) = R 2 . Without loss of generality, we may assume that N = N 1 × M 2 for some proper submodule N 1 of M 1. Now, we will show that N 1 is a classical prime submodule of M 1. Let abm ∈ N 1 for some a, b ∈ R 1 and m ∈ M 1. Since Ann R (M 2) is not unique maximal ideal of R 2, there exists a nonunit t ∉ Ann R (M 2). Then there exists n ∈ M 2 such that tn ≠ 0. Then we have 0 M ≠ (abm, tn) = (a, 1)(1, t)(b, 1)(m, n) ∈ N. As N is a weakly classical 1-absorbing prime submodule of M, we have either (a, 1)(1, t)(m, n) ∈ N or (b, 1)(m, n) ∈ N which implies that am ∈ N 1 or bm ∈ N 1. Thus, N 1 is a classical prime submodule of M 1.

(2) ⇒ (3): Follows from [8], Theorem 2.35].

(3) ⇒ (4) ⇒ (1): Clear. □

Theorem 13.

Let R be a um-ring and M be an R-module.

(1) If F is a flat R-module and N is a weakly classical 1-absorbing prime submodule of M such that F ⊗ N ≠ F ⊗ M, then F ⊗ N is a weakly classical 1-absorbing prime submodule of F ⊗ M.

(2) Suppose that F is a faithfully flat R-module. Then N is a weakly classical 1-absorbing prime submodule of M if and only if F ⊗ N is a weakly classical 1-absorbing prime submodule of F ⊗ M.

Proof.

(1): Let a, b, c ∈ R be nonunits. Then we get by Theorem 9, either N : M a b c = 0 : M a b c or N : M a b c = N : M a b or N : M a b c = N : M c . Assume that N : M a b c = 0 : M a b c . Then by [7], Lemma 3.2], we have F ⊗ N : F ⊗ M a b c = F ⊗ N : M a b c = F ⊗ 0 : M a b c = F ⊗ 0 : F ⊗ M a b c = 0 : F ⊗ M a b c . Now, suppose that N : M a b c = N : M a b . Again by [7], Lemma 3.2], we have F ⊗ N : F ⊗ M a b c = F ⊗ N : M a b c = F ⊗ N : M a b = F ⊗ N : F ⊗ M a b . Similarly, we can show that if N : M a b c = N : M c , then F ⊗ N : F ⊗ M a b c = F ⊗ N : F ⊗ M c . Consequently, by Theorem 9, we deduce that F ⊗ N is a weakly classical 1-absorbing prime submodule of F ⊗ M.

(2): Let N be a weakly classical 1-absorbing prime submodule of M and assume F ⊗ N = F ⊗ M. Then 0 → F ⊗ N → ⊆ F ⊗ M → 0 is an exact sequence. Since F is a faithfully flat R-module, 0 → N → M → 0 is an exact sequence. So N = M, which is a contradiction. Thus we have F ⊗ N ≠ F ⊗ M. Then F ⊗ N is a weakly classical 1-absorbing prime submodule of F ⊗ M by (1). Now for the converse, let F ⊗ N be a weakly classical 1-absorbing prime submodule of F ⊗ M. Then we have F ⊗ N ≠ F ⊗ M and so N ≠ M. Let a, b, c ∈ R be nonunits. Then F ⊗ N : F ⊗ M a b c = 0 : F ⊗ M a b c or F ⊗ N : F ⊗ M a b c = F ⊗ N : F ⊗ M a b or F ⊗ N : F ⊗ M a b c = F ⊗ N : F ⊗ M c by Theorem 9. Assume that F ⊗ N : F ⊗ M a b c = 0 : F ⊗ M a b c . Hence F ⊗ N : M a b c = F ⊗ N : F ⊗ M a b c = 0 : F ⊗ M a b c = F ⊗ 0 : M a b c . So 0 → F ⊗ 0 : M a b c → ⊆ F ⊗ N : M a b c → 0 is exact sequence. Since F is a faithfully flat R-module 0 → 0 : M a b c → ⊆ N : M a b c → 0 is an exact sequence which implies that N : M a b c = 0 : M a b c . By a similar argument, we can deduce that if F ⊗ N : F ⊗ M a b c = F ⊗ N : F ⊗ M a b or F ⊗ N : F ⊗ M a b c = F ⊗ N : F ⊗ M c , then N : M a b c = N : M a b or N : M a b c = N : M c . Consequently, N is a weakly classical 1-absorbing prime submodule of M by Theorem 9. □

Corollary 6.

Let R be an um-ring, M be an R-module, and x be an indeterminate. If N is a weakly classical 1-absorbing prime submodule of M, then N x is a weakly classical 1-absorbing prime submodule of M x .

Proof.

Assume that N is a weakly classical 1-absorbing prime submodule of M. Notice that R x is a flat R-module. So by Theorem 13, N x ≃ R x ⊗ N is a weakly classical 1-absorbing prime submodule of M [ x ] ≃ R x ⊗ M . □

Now, we investigate the modules over which every proper ideal is weakly classical 1-absorbing prime.

Theorem 14.

Let M be an R-module. The following statements are satisfied.

(1) If every proper submodule of M is a weakly classical 1-absorbing prime. Then Jac(R)3 M = 0.

(2) Suppose that ( R , m ) is a local ring. Then every proper submodule of M is a weakly classical 1-absorbing prime if and only if m 3 M = 0 .

Proof.

(1): Suppose that every proper submodule of M is a weakly classical 1-absorbing prime. Now, we will show that Jac(R)3 M = 0. Suppose to the contrary. Then there exist x, y, z ∈ Jac(R) and m ∈ M such that xyzm ≠ 0. Now, put N = Rxyzm. Since 0 ≠ xyzm ∈ N and N is a weakly classical 1-absorbing prime submodule, we have either xym ∈ N or zm ∈ N. Then there exists a ∈ R such that xym = axyzm or zm = axyzm. In the first case, we have (1 − az)xym = 0. Since 1 − az is unit, we conclude that xym = 0. In the second case, one can similarly show that zm = 0. Thus, we conclude, xyzm = 0 which is a contradiction. Hence, Jac(R)3 M = 0.

(2): Suppose that ( R , m ) is a local ring. If every proper submodule of M is weakly classical 1-absorbing prime, then by (1), m 3 M = 0 . For the converse, note that xyz = 0 for every nonunits x, y, z ∈ R. In this case, every proper submodule of M is trivially weakly classical 1-absorbing prime submodule. □

The following example shows that the converse of Theorem 14 is not true in general.

Example 7.

Consider the R = Z -module M = Z × Z and the submodule N = 2 Z × 3 Z . Then Jac(R) = (0) and this gives Jac(R)3 M = (0). Since (0, 0) ≠ 2 ⋅ 5 ⋅ 3 ⋅ (1, 1) = (30, 30) ∈ N, but 2 ⋅ 5 ⋅ (1, 1) = (10, 10) ∉ N and 3 ⋅ (1, 1) = (3, 3) ∉ N, it follows that N is not a weakly classical 1-absorbing prime submodule of M.

Corresponding author: Zeynep Yılmaz, Department of Mathematics, Yildiz Technical University, Istanbul, Türkiye, E-mail: 

Acknowledgments

The authors are grateful for the reviewer’s valuable comments that improved the manuscript.

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. All authors contributed equally to the manuscript.

  4. Conflict of interest: Authors state no conflict of interest.

  5. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  6. Research funding: Authors state no funding involved.

  7. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

References

[1] M. Behboodi, On weakly prime radical of modules and semi-compatible modules, Acta Math. Hungar. 113 (2006), no. 3, 243–254, https://doi.org/10.1007/s10474-006-0097-6.Suche in Google Scholar

[2] A. Y. Darani and F. Soheilnia, 2-absorbing and weakly 2-absorbing submodules, Thai J. Math. 9 (2011), no. 3, 577–584.Suche in Google Scholar

[3] A. Y. Darani and F. Soheilnia, On n-absorbing submodules, Math. Commun. 17 (2012), no. 2, 547–557.Suche in Google Scholar

[4] D. D. Anderson, T. Arabaci, Ü. Tekir, and S. Koç, On S-multiplication modules, Comm. Algebra 48 (2020), no. 8, 3398–3407, https://doi.org/10.1080/00927872.2020.1737873.Suche in Google Scholar

[5] M. Behboodi, A generalization of Baer’s lower nilradical for modules, J. Algebra Appl. 6 (2007), no. 2, 337–353, https://doi.org/10.1142/S0219498807002284.Suche in Google Scholar

[6] M. Behboodi and H. Koohy, Weakly prime modules, Vietnam J. Math. 32 (2004), no. 2, 185–195.Suche in Google Scholar

[7] A. Azizi, Weakly prime submodules and prime submodules, Glasg. Math. J. 48 (2006), no. 2, 343–346, https://doi.org/10.1017/S0017089506003119.Suche in Google Scholar

[8] H. Mostafanasab, Ü. Tekir, and K. H. Oral, Weakly classical prime submodules, Kyungpook Math. J. 56 (2016), no. 4, 1085–1101, https://doi.org/10.5666/KMJ.2016.56.4.1085.Suche in Google Scholar

[9] A. Yassine, M. J. Nikmehr, and R. Nikandish, On 1-absorbing prime ideals of commutative rings, J. Algebra Appl. 20 (2021), no. 10, 2150175, https://doi.org/10.1142/S0219498821501759.Suche in Google Scholar

[10] E. Yildiz, Ü. Tekir, and S. Koc, On ϕ-1-absorbing prime ideals, Beitr. Algebra Geom. 62 (2021), no. 4, 907–918, https://doi.org/10.1007/s13366-020-00557-w.Suche in Google Scholar

[11] E. M. Bouba, M. Tamekkante, Ü. Tekir, and S. Koç, Notes on 1-absorbing prime ideals, C. R. Acad. Bulgare Sci. 75 (2022), no. 5, 631–639, https://doi.org/10.7546/CRABS.2022.05.01.Suche in Google Scholar

[12] G. Ulucak, S. Koç, and Ü. Tekir, On n-1-absorbing prime ideals, J. Algebra Appl. 22 (2023), no. 10, 2350207, https://doi.org/10.1142/S0219498823502079.Suche in Google Scholar

[13] S. Koç, Ü. Tekir, and E. Yıldız, On weakly 1-absorbing prime ideals, Ric. Mat. 72 (2023), no. 2, 723–738, https://doi.org/10.1007/s11587-020-00550-4.Suche in Google Scholar

[14] R. Y. Sharp, Steps in Commutative Algebra, Cambridge University Press, Cambridge, 2000.10.1017/CBO9780511623684Suche in Google Scholar

[15] A. Barnard, Multiplication modules, J. Algebra 71 (1981), no. 1, 174–178, https://doi.org/10.1016/0021-8693(81)90112-5.Suche in Google Scholar

[16] R. Ameri, On the prime submodules of multiplication modules, Int. J. Math. Math. Sci. 27 (2003), 1715–1724, https://doi.org/10.1155/S0161171203202180.Suche in Google Scholar

[17] N. Groenewald, 1-absorbing and weakly 1-absorbing prime submodules of a module over a noncommutative ring, Afr. Mat. 34 (2023), no. 3, 54, https://doi.org/10.1007/s13370-023-01094-0.Suche in Google Scholar

[18] M. Ebrahimpour and F. Mirzaee, On ϕ-semiprime submodules, J. Korean Math. Soc. 54 (2017), no. 4, 1099–1108, https://doi.org/10.4134/JKMS.j160075.Suche in Google Scholar

[19] T. K. Lee and Y. Zhou, Reduced modules, in: Rings, Modules, Algebras, and Abelian Groups, Lecture Notes in Pure and Applied Mathematics, vol. 236, Marcel Dekker, New York, 2004, pp. 365–378.10.1201/9780824750817.ch29Suche in Google Scholar

[20] P. Quartararo and H. S. Butts, Finite unions of ideals and modules, Proc. Amer. Math. Soc. 52 (1975), no. 1, 91–96, DOI https://doi.org/10.1090/s0002-9939-1975-0382249-5.Suche in Google Scholar

[21] M. D. Larsen and P. J. McCarthy, Multiplicative Theory of Ideals, Pure and Applied Mathematics, vol. 43, Academic Press, New York, 1971.Suche in Google Scholar

[22] Z. Yılmaz, B. A. Ersoy, Ü. Tekir, S. Koç, and S. Onar, On classical 1-absorbing prime submodules, Indian J. Pure Appl. Math. (2025), 1–13, https://doi.org/10.1007/s13226-025-00783-9.Suche in Google Scholar

[23] M. M. Ali, Idempotent and nilpotent submodules of multiplication modules, Comm. Algebra 36 (2008), no. 12, 4620–4642, https://doi.org/10.1080/00927870802186805.Suche in Google Scholar

[24] P. F. Smith, Some remarks on multiplication modules, Arch. Math. 50 (1988), 223–235, https://doi.org/10.1007/BF01187738.Suche in Google Scholar
Received: 2025-04-02
Accepted: 2025-10-24
Published Online: 2025-11-24

© 2025 the author(s), published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2025-0215
Schlagwörter für diesen Artikel
weakly classical prime submodules; weakly classical 1-absorbing prime submodules; weakly classical 2-absorbing submodules
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Artikel in diesem Heft

  1. Special Issue on Contemporary Developments in Graphs Defined on Algebraic Structures
  2. Forbidden subgraphs of TI-power graphs of finite groups
  3. Finite group with some c#-normal and S-quasinormally embedded subgroups
  4. Classifying cubic symmetric graphs of order 88p and 88p 2
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  8. Algebra
  9. Classes of modules closed under projective covers
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  20. Kodaira-Spencer maps for elliptic orbispheres as isomorphisms of Frobenius algebras
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Heruntergeladen am 6.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/math-2025-0215/html
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