let $r$ be a commutative ring. the purpose of this article is to introduce a new class of ideals of r called weakly irreducible ideals. this class could be a generalization of the families quasi-primary ideals and strongly irreducible ideals. the relationships between the notions primary, quasi-primary, weakly irreducible, strongly irreducible and irreducible ideals, in different rings, has bee...

in this paper we introduce the notions of uniformly quasi-primary ideals and uniformly classical quasi-primary submodules that generalize the concepts of uniformly primary ideals and uniformly classical primary submodules; respectively. several characterizations of classical quasi-primary and uniformly classical quasi-primary submodules are given. then we investigate for a ring $r$, when any fi...

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...

notions of strongly regular, regular and left(right) regular $gamma$−semigroupsare introduced. equivalent conditions are obtained through fuzzy notion for a$gamma$−semigroup to be either strongly regular or regular or left regular.

In this paper, we define the new notion of quasi-prime ideal which generalizes at once both prime ideal and primary ideal notions. Then a natural topology on the set of quasi-prime ideals of a ring is introduced which admits the Zariski topology as a subspace topology. The basic properties of the quasi-prime spectrum are studied and several interesting results are obtained. Specially, it is pro...

in this paper we introduce the notion of classical quasi-primary submodules that generalizes the concept of classical primary submodules. then, we investigate decomposition and minimal decomposition into classical quasi-primary submodules. in particular, existence and uniqueness of classical quasi-primary decompositions in finitely generated modules over noetherian rings are proved. moreover, w...

Let $R$ be a commutative ring. The purpose of this article is to introduce a new class of ideals of R called weakly irreducible ideals. This class could be a generalization of the families quasi-primary ideals and strongly irreducible ideals. The relationships between the notions primary, quasi-primary, weakly irreducible, strongly irreducible and irreducible ideals, in different rings, has bee...

the main objective of this study is to swing krull intersection theorem in primary decomposition of rings and modules to the primary decomposition of soft rings and soft modules. to fulfill this aim several notions like soft prime ideals, soft maximal ideals, soft primary ideals, and soft radical ideals are introduced for a soft ring over a given unitary commutative ring. consequently, the p...

let r be a commutative ring with identity. let n and k be two submodules of a multiplication r-module m. thenn=im and k=jm for some ideals i and j of r. the product of n and k denoted by nk is defined by nk=ijm. inthis paper we characterize some particular cases of multiplication modules by using the product of submodules.

the rings considered in this article are commutative with identity $1neq 0$. by a proper ideal of a ring $r$,  we mean an ideal $i$ of $r$ such that $ineq r$.  we say that a proper ideal $i$ of a ring $r$ is a  maximal non-prime ideal if $i$ is not a prime ideal of $r$ but any proper ideal $a$ of $r$ with $ isubseteq a$ and $ineq a$ is a prime ideal. that is, among all the proper ideals of $r$,...