On Graded Weakly S-Prime Ideals

Let R be a commutative graded ring with unity, S be a multiplicative subset of homogeneous elements of R and P be a graded ideal of R such that P\bigcap S=\emptyset In this article, we introduce several results concerning graded S-prime ideals. Then we introduce the concept of graded weakly S-prime ideals which is a generalization of graded weakly prime ideals. We say that P is a graded weakly S-prime ideal of R if there exists s\in S such that for all x, y\in h(R), if 0\neq xy\in P, then sx\in P or sy\in P. We show that graded weakly S-prime ideals have many acquaintance properties to these of graded weakly prime ideals.

On weakly S-prime ideals of commutative rings

Let R be a commutative ring with identity and S be a multiplicative subset of R. In this paper, we introduce the concept of weakly S-prime ideals which is a generalization of weakly prime ideals. Let P be an ideal of R disjoint with S. We say that P is a weakly S-prime ideal of R if there exists an s ∈ S such that, for all a, b ∈ R, if 0 ≠ ab ∈ P, then sa ∈ P or sb ∈ P. We show that weakly S-prime ideals have many analog properties to these of weakly prime ideals. We also use this new class of ideals to characterize S-Noetherian rings and S-principal ideal rings.

On Graded 2-Prime Ideals

The purpose of this paper is to introduce the concept of graded 2-prime ideals as a new generalization of graded prime ideals. We show that graded 2-prime ideals and graded semi-prime ideals are different. Furthermore, we show that graded 2-prime ideals and graded weakly prime ideals are also different. Several properties of graded 2-prime ideals are investigated. We study graded rings in which every graded 2-prime ideal is graded prime, we call such a graded ring a graded 2-P-ring. Moreover, we introduce the concept of graded semi-primary ideals, and show that graded 2-prime ideals and graded semi-primary ideals are different concepts. In fact, we show that graded semi-primary, graded 2-prime and graded primary ideals are equivalent over Z-graded principal ideal domain.

Almost prime submodules

In this paper, we introduce the notion of almost prime submodules of a given right [Formula: see text]-module and study some properties of them as a generalization of weakly prime ideals in associative rings. We will also investigate modules in which every fully invariant submodule is almost prime.

On ordered hypersemigroups with idempotent ideals, prime and weakly prime ideals

Some well known results on ordered semigroups are examined in case of ordered hypersemigroups. Following the paper in Semigroup Forum 44 (1992), 341--346, we prove the following: The ideals of an ordered hypergroupoid$H$ are idempotent if and only if for any two ideals $A$ and $B$ of $H$, we have $A\cap B=(A*B]$. Let now $H$ be an ordered hypersemigroup. Then, the ideals of $H$ are idempotent if and only if $H$ is semisimple. The ideals of $H$ are weakly prime if and only if they are idempotent and they form a chain. The ideals of $H$ are prime if and only if they form a chain and $H$ is intra-regular. The paper serves as an example to show how we pass from ordered semigroups to ordered hypersemigroups.

On quasi-prime, weakly quasi-prime left ideals in ordered-Γ-semigroups

In this paper we obtain and establish some important results in ordered Γ-semigroups extending and generalizing those for semigroups given in [PETRICH, M.: Introduction to Semigroups, Merill, Columbus, 1973] and for ordered semigroups from [KEHAYOPULU, N.: On weakly prime ideals of ordered semigroups, Math. Japon. 35 (1990), 1051–1056], [KEHAYOPULU, N.: On prime, weakly prime ideals in ordered semigroups, Semigroup Forum 44 (1992), 341–346] and [XIE, X. Y.—WU, M. F.: On quasi-prime, weakly quasi-prime left ideals in ordered semigroups, PU.M.A. 6 (1995), 105–120]. We introduce and give some characterizations about the quasi-prime and weakly quasi-prime left ideals of ordered-Γ-semigroups. We also introduce the concept of weakly m-systems in ordered Γ-semigroups and give some characterizations of the quasi-prime and weakly quasi-prime left ideals by weakly m-systems.