let r be a commutative ring with identity and m be a unitary r-module. let : s(m) −! s(m) [ {;} be a function, where s(m) is the set of submodules ofm. suppose n  2 is a positive integer. a proper submodule p of m is called(n − 1, n) − -prime, if whenever a1, . . . , an−1 2 r and x 2 m and a1 . . . an−1x 2p(p), then there exists i 2 {1, . . . , n − 1} such that a1 . . . ai−1ai+1 . . . an−1x...

In this article, we develop the tool of saturation in the context of primary-like submodules of modules. We are particularly interested in relationships among the saturation of a primary-like submodule satisfying the primeful property and its radical. Furthermore, we provide sufficient conditions involving saturation and torsion arguments under which the radical of such a submodule is prime.

Let R be a commutative ring with identity and let M be a torsion free R-module. Several characterizations of distributive modules are investigated. Indeed, among other equivalent conditions, we prove that M is distributive if and only if any primal submodule of M is irreducible, and, if and only if each submodule of M can be represented as an intersection of irreducible isolated components. MSC...

Let Γ=(V,E) be a graph and ‎W_(‎a)={w_1,…,w_k } be a subset of the vertices of Γ and v be a vertex of it. The k-vector r_2 (v∣ W_a)=(a_Γ (v,w_1),‎…‎ ,a_Γ (v,w_k)) is the adjacency representation of v with respect to W in which a_Γ (v,w_i )=min{2,d_Γ (v,w_i )} and d_Γ (v,w_i ) is the distance between v and w_i in Γ. W_a is called as an adjacency resolving set for Γ if distinct vertices of ...

Let R be a commutative ring with identity and M be a unitary R-module. Let : S(M) −! S(M) [ {;} be a function, where S(M) is the set of submodules ofM. Suppose n 2 is a positive integer. A proper submodule P of M is called(n − 1, n) − -prime, if whenever a1, . . . , an−1 2 R and x 2 M and a1 . . . an−1x 2P(P), then there exists i 2 {1, . . . , n − 1} such that a1 . . . ai−1ai+1 . . . an−1x 2 P...

Let $G$ be a group with identity $e$. $R$ commutative $G$-graded ring non-zero identity, $S\subseteq h(R)$ multiplicatively closed subset of and $M$ graded $R$-module. In this article, we introduce study the concept $S$-1-absorbing prime submodules. A submodule $N$ $(N:_{R}M)\cap S=\emptyset$ is said to prime, if there exists an $s_{g}\in S$ such that whenever $a_{h}b_{h'}m_{k}\in N$, then eith...

let $n$ be a submodule of a module $m$ and a minimal primary decomposition of $n$ is known‎. ‎a formula to compute baer's lower nilradical of $n$ is given‎. ‎the relations between classical prime submodules and their nilradicals are investigated‎. ‎some situations in which semiprime submodules can be written as finite intersection of classical prime submodule are stated‎.

let $n$ be a submodule of a module $m$ and a minimal primary decomposition of $n$ is known‎. ‎a formula to compute baer's lower nilradical of $n$ is given‎. ‎the relations between classical prime submodules and their nilradicals are investigated‎. ‎some situations in which semiprime submodules can be written as finite intersection of classical prime submodule are stated‎.

in this paper we investigate decompositions of submodules in modules over a proufer domain into intersections of quasi-primary and classical quasi-primary submodules. in particular, existence and uniqueness of quasi-primary decompositions in modules over a proufer domain of finite character are proved. proufer domain; primary submodule; quasi-primary submodule; classical quasi-primary; decomposi...