Related papers: $\phi$-prime submodules

Let $\mathfrak F$ be a formation and let $G$ be a group. A subgroup $H$ of $G$ is $\mathrm{K}\mathfrak F$-subnormal (submodular) in $G$ if there is a subgroup chain $H=H_0\le \ H_1 \le \ \ldots \le H_i \leq H_{i+1}\le \ldots \le \ H_n=G$…

Let $R$ be a commutative ring, $M$ an $R$-module and $\varphi_a$ be the endomorphism of $M$ given by right multiplication by $a\in R$. We say that $M$ is {\it weakly-morphic} if $M/\varphi_a(M)\cong \ker(\varphi_a)$ as $R$-modules for every…

Suppose $R$ is a commutative ring and $G$ is a group acting on a set $W$. We consider the $RG$-module $RW$ in the case where $G$ is the automorphism group of an $\omega$-categorical structure $M$ and $W$ is, for example, $M^n$ (for $n \in…

Let $G$ be a finite group and $H$ be a subgroup of $G$. Then $H$ is called a weakly $S\Phi$-supplemented subgroup of $G$, if there exists a subgroup $T$ of $G$ such that $G =HT$ and $H \cap T \leq \Phi (H) H_{sG}$, where $H_{sG}$ denotes…

Given a multiplicative subset $S$ in a commutative ring $R$, we consider $S$-weakly cotorsion and $S$-strongly flat $R$-modules, and show that all $R$-modules have $S$-strongly flat covers if and only if all flat $R$-modules are…

Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and $\Pi$ a non-empty subset of the set $\sigma$. A set ${\cal H}$ of subgroups of a finite group $G$ is said to be a \emph{ complete Hall $\Pi…

In this paper, we associate a new topology to a nonzero unital module $M$ over a commutative $R$, which is called Golomb topology of the $R$-module $M$. Let $M\ $be an\ $R$-module and $B_{M}$ be the family of coprime cosets $\{m+N\}$ where…

A submodule $W$ of a p-primary module $M$ of bounded order is known to be regular if $W$ and $M$ have simultaneous bases. In this paper we derive necessary and sufficient conditions for regularity of a submodule.

In this article, we introduce a generalization of the concept of graded $r$-ideals in graded commutative rings with nonzero unity. Let $G$ be a group, $R$ be a $G$-graded commutative ring with nonzero unity and $GI(R)$ be the set of all…

In this paper, we introduce and study the class of $\phi$-$w$-flat modules which are generalizations of both $\phi$-flat modules and $w$-flat modules. The $\phi$-$w$-weak global dimension $\phi$-$w$-w.gl.dim$(R)$ of a strongly $\phi$-ring…

Let R be a commutative ring with identity, S be a multiplicatively closed subset of R, and let M be an R-module. In this paper, we introduce the notion of S-2-absorbing second submodules of M as a generalization of S-second submodules and…

In this paper, we introduce and study the notion of strongly $\phi$-$w$-flat modules. The $\phi$-$w$-weak global dimension $\phi$-$w$-w.gl.dim$(R)$ of an NP-ring $R$ is also introduced and studied. We characterize $\phi$-\Prufer\…

Let R be a commutative ring with unity and M be an R- module In this paper we introduce semi n- absorbing and (k, n)-closed submodules of modules over commutative rings, and investigate their basic properties.

We study the so-called closed and splitting subsemimodules and submodules of a given semimodule or module, respectively. We describe lattices of subsemimodules and of closed subsemimodules and posets of splitting subsemimodules and…

For commutative rings with identity, we introduce and study the concept of semi $r$-ideals which is a kind of generalization of both $r$-ideals and semiprime ideals. A proper ideal $I$ of a commutative ring $R$ is called semi $r$-ideal if…

Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring, $M$ a graded $R$-module and $A\subseteq h(R)$ a multiplicatively closed subset of $R$. In this paper, we introduce the concept of graded $A$-2-absorbing…

Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. An $R$-module $M$ is said to be a uniformly $S$-Artinian ($u$-$S$-Artinian for abbreviation) module if there is $s\in S$ such that any descending chain of…

In this paper, we introduce a new concept in Nil-semicommutative modules and present it as an extension of Nil-semicommutative rings to modules. We prove that the class of Nil-semicommutative modules is contained in the class of Weakly…

The purpose of the present paper is to continue the study of modules cofinite and weakly cofinite with respect to an ideal $\frak a$ of a Noetherian ring $R$. It is shown that an $R$-module $M$ is cofinite with respect to $\frak a$, if and…

Let $R$ be a commutative ring with identity. In this paper, we introduce the concept of weakly $1$-absorbing prime ideals which is a generalization of weakly prime ideals. A proper ideal $I$ of $R$ is called weakly $1$-absorbing prime if…