arXiv subjects

Alaa Abouhalaka

Publications and source records attributed to Alaa Abouhalaka.

4 recordsLinked to original sources

arXiv2024

S-J-Ideals: A Study in Commutative and Noncommutative Rings

In this paper, we introduce the concept of S-J-ideals in both commutative and noncommutative rings. For a commutative ring R and a multiplicatively closed subset S, we show that many properties of J-ideals apply to S-J-ideals and examine their characteristics in various ring constructions, such as homomorphic image rings, quotient rings, cartesian product rings, polynomial rings, power series rings, idealization rings, and amalgamation rings. In noncommutative rings, where S is an m-system, we define right S-J-ideals. We demonstrate the equivalence of S-J-ideals and right S-J-ideals in commutative rings with identity and provide examples to illustrate the connections between right S-prime ideals and J-ideals.

math.RA↗

arXiv2024

Generalizations of almost prime and right $S$-prime ideals in noncommutative rings

Let $R$ be a noncommutative ring, and let $S$ be an $m$-system of $R$. In this paper, we give more results on the concept of almost prime (right) ideals, that were introduced by the first two authors, especially in (right) $S$-unital rings, local rings, and decomposable rings. In addition, we introduce the concept of almost right $S$-prime ideals, and we show how some findings regarding almost prime ideals can be derived as consequences of almost right $S$-prime ideals. Besides, we show how almost right $S$-prime ideals behave in related rings such as homomorphic images, quotient rings, and decomposable rings. Finally, we construct almost right $S$-prime ideals using the Nagata method of idealization.

math.RA↗

arXiv2024

S-Prime Right Submodules and an S-Version of Prime Avoidance

Let S be an m-system of a ring R, and P a submodule of a right R-module M. This paper, presents the notion of S-prime submodule and provides some properties and equivalent definitions. We define S-multiplication right module, and prove that in multiplication (S-multiplication) right R-module M, the ideal (P :_R M) is a right S-prime ideal of R if and only if P is an S-prime submodule of M. Moreover, we give an S-version of prime avoidance lemma. Furthermore, we define S-finite and S-Noetherian right modules following the definitions in [1]. We prove that a multiplication finitely generated right R-module M is S-Noetherian if (N :_R M) is an S-prime ideal of R, for all submodules N of M. In addition, we give some examples of right S-Noetherian rings.

math.RA↗

arXiv2022

Almost prime ideals in noncommutative rings

A proper ideal $P$ of a commutative ring with identity is an almost prime ideal if $ab \in P{\setminus}P^2$ implies $a \in P$ or $b \in P$. In this paper we define almost prime ideals of a noncommutative ring, and provide some equivalent definitions. We also examine some cases such that all right ideals of a noncommutative ring are almost prime right ideals.

math.RA↗