On weakly S-prime submodules

Abstract

Let R be a commutative ring with a non-zero identity, S be a multiplicatively closed subset of R and M be a unital R-module. In this paper, we define a submodule N of M with (N : R M ) ∩ S = φ to be weakly S-prime if there exists s ∈ S such that whenever a ∈ R and m ∈ M with 0 = am ∈ N , then either sa ∈ (N : R M ) or sm ∈ N . Many properties, examples and characterizations of weakly S-prime submodules are introduced, especially in multiplication modules. Moreover, we investigate the behavior of this structure under module homomorphisms, localizations, quotient modules, cartesian product and idealizations. Finally, we define two kinds of submodules of the amalgamation module along an ideal and investigate conditions under which they are weakly S-prime.

On Weakly S-2-Absorbing Submodules

“…Of course a proper submodule P of M is called prime if am ∈ P for a ∈ R and m ∈ M implies a ∈ (P : R M) or m ∈ P where (P : R M) = {r ∈ R : r M ⊆ P }. Several generalizations of these concepts have been studied extensively by many authors [9], [13], [6], [16], [3], [11], [14], [5].…”

“…After that, the concept of weakly S-prime submodule was introduced as a generalization of S-prime submodules in [11]. Here, for a multiplicatively closed subset S of R, they called a submodule P of an R-module M with (P : R M) ∩ S = ∅ a weakly S-prime submodule if there exists s ∈ S such that for a ∈ R and m ∈ M, if 0 = am ∈ P then either sa ∈ (P : R M) or sm ∈ P .…”

On gw-Prime Submodules

“…In [2], Rashid Abu-Dawwas, Khaldoun Al-Zoubi introduced the concept of graded weakly classical prime submodules and they gave some properties of these submodules. In [3], Hani A. Khashan and Ece Yetkin Celikel introduced a new type of weakly prime submodules which they called weakly 𝑆 −prime submodules and gave many properties and characterizations of them in multiplication modules. In [4] Emel Aslankarayigit Ugurlu proved some results concerning Sprime and S-weakly prime submodules.…”