arXiv subjects

Hani A. Khashan

Publications and source records attributed to Hani A. Khashan.

7 recordsLinked to original sources

arXiv2022

Semi r-ideals of commutative rings

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 whenever $a^{2}\in I$ and $Ann_{R}(a)=0$, then $a\in I$. Several properties and characterizations of this class of ideals are determined. In particular, we investigate semi $r$-ideal under various contexts of constructions such as direct products, localizations, homomorphic images, idealizations and amalagamations rings. We extend semi $r$-ideals of rings to semi $r$-submodules of modules and clarify some of their properties. Moreover, we define submodules satisfying the $D$-annihilator condition and justify when they are semi $r$-submodules.

math.AC↗

arXiv2022

On Weakly S-primary Submodules

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 $(N:_{R}M)\cap S=\emptyset$ to be weakly $S$-primary if there exists $s\in S$ such that whenever $a\in R$ and $m\in M$ with $0\neq am\in N$, then either $sa\in\sqrt{(N:_{R}M)}$ or $sm\in N$. We present various properties and characterizations of this concept (especially in finitely generated faithful multiplication modules). Moreover, the behavior of this structure under module homomorphisms, localizations, quotient modules, cartesian product and idealizations is investigated. Finally, we determine some conditions under which two kinds of submodules of the amalgamation module along an ideal are weakly $S$-primary.

math.AC↗

arXiv2021

On weakly S-prime submodules

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 $(N:_{R}M)\cap S=ϕ$ to be weakly $S$-prime if there exists $s\in S$ such that whenever $a\in R$ and $m\in M$ with $0\neq am\in N$, then either $sa\in(N:_{R}M)$ or $sm\in N$. 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.

math.AC↗

arXiv2021

Weakly J-submodules of Modules over Commutative Rings

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. By $J(R),$ we denote the Jacobson radical of $R$. The purpose of this paper is to introduce the concept of weakly $J$-submodules generalizing $J$-submodules. We call a proper submodule $N$ of $M$ a weakly $J$-submodule if whenever $0\neq rm\in N$ for $r\in R$ and $m\in M$, then $r\in(J(R)M:M)$ or $m\in N.$ Various properties and characterizations of weakly $J$-submodules are investigated especially in the case of multiplication modules.

math.AC↗

arXiv2021

Weakly J-ideals of Commutative Rings

Let $R$ be a commutative ring with non-zero identity. In this paper, we introduce the concept of weakly $J$-ideals as a new generalization of $J$-ideals. We call a proper ideal $I$ of a ring $R$ a weakly $J$-ideal if whenever $a,b\in R$ with $0\neq ab\in I$ and $a\notin J(R)$, then $a\in I$. Many of the basic properties and characterizations of this concept are studied. We investigate weakly $J$-ideals under various contexts of constructions such as direct products, localizations, homomorphic images. Moreover, a number of examples and results on weakly $J$-ideals are discussed. Finally, the third section is devoted to the characterizations of these constructions in an amagamated ring along an ideal.

math.AC↗

arXiv2021

Quasi J-ideals of Commutative Rings

Let $R$ be a commutative ring with identity. In this paper, we introduce the concept of quasi $J$-ideal which is a generalization of $J$-ideal. A proper ideal of $R$ is called a quasi $J$-ideal if its radical is a $J$-ideal. Many characterizations of quasi $J$-ideals in some special rings are obtained. We characterize rings in which every proper ideal is quasi $J$-ideal. Further, as a generalization of presimplifiable rings, we define the notion of quasi presimplifiable rings. We call a ring $R$ a quasi presimplifiable ring if whenever $a,b\in R$ and $a=ab$, then either $a$ is a nilpotent or $b$ is a unit. It is shown that a proper ideal $I$ that is contained in the Jacobson radical is a quasi $J$-ideal (resp. $J$-ideal) if and only if $R/I$ is a quasi presimplifiable (resp. presimplifiable) ring.

math.AC↗

arXiv2021

Quasi J-submodules

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. The aim of this paper is to extend the notion of quasi $J$-ideals of commutative rings to quasi $J$-submodules of modules. We call a proper submodule $N$ of $M$ a quasi $J$-submodule if whenever $r\in R$ and $m\in M$ such that $rm\in N$ and $r\notin(J(R)M:M)$, then $m\in M$-$rad(N)$. We present various properties and characterizations of this concept (especially in finitely generated faithful multiplication modules). Furthermore, we provide new classes of modules generalizing presimplifiable modules and justify their relation with (quasi) $J$-submodules. Finally, for a submodule $N$ of $M$ and an ideal $I$ of $R$, we characterize the quasi $J$-ideals of the idealization ring $R(+)M$.

math.AC↗