Graded 1-absorbing prime ideals of graded commutative rings

Let [Formula: see text] be a group with identity [Formula: see text] and [Formula: see text] be [Formula: see text]-graded commutative ring with [Formula: see text] In this paper, we introduce and study the graded versions of 1-absorbing prime ideal. We give some properties and characterizations of these ideals in graded ring, and we give a characterization of graded 1-absorbing ideal the idealization [Formula: see text]

Strong regularity of smash products associated with G-set gradings

For a group G, a G-graded ring R and a finite left G-set A, we study the strong regularity of the smash product of R and A.

Characterizations of Graded Rings over Which Every Graded semi-primary Ideal is Graded 1-Absorbing Primary

Let G be a group with identity e and R be a commutative G-graded ring with nonzero unity 1. Graded semi-primary and graded 1-absorbing primary ideals have been investigated and examined by several authors as generalizations of graded primary ideals. However, these three concepts are different. In this article, we character­ ize graded rings over which every graded semi-primary ideal is graded 1-absorbing primary and graded rings over which every graded 1-absorbing primary ideal is graded primary.

On Graded S-Primary 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⋂S=∅. In this article, we introduce the concept of graded S-primary ideals which is a generalization of graded primary ideals. We say that P is a graded S-primary ideal of R if there exists s∈S such that for all x,y∈h(R), if xy∈P, then sx∈P or sy∈Grad(P) (the graded radical of P). We investigate some basic properties of graded S-primary ideals.

Characterizations of Graded Rings Over Which Every Graded Semi-primary Ideal is Graded 1-absorbing Primary

Let $G$ be a group with identity $e$ and $R$ be a commutative $G$-graded ring with nonzero unity $1$. Graded semi-primary and graded $1$-absorbing primary ideals have been investigated and examined by several authors as generalizations of graded primary ideals. However, these three concepts are different. In this article, we characterize graded rings over which every graded semi-primary ideal is graded $1$-absorbing primary and graded rings over which every graded $1$-absorbing primary ideal is graded primary.

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.

Generalizations of graded second submodules

Let G be a group with identity e. Let R be a graded ring, I a graded ideal of R and M be a G-graded R-module. Let ψ: Sgr(M) → S gr(M) ∪ {∅} be a function, where Sgr(M) denote the set of all graded submodules of M. In this article, we introduce and study the concepts of graded ψ -second submodules and graded I-second submodules of a graded R-module which are generalizations of graded second submodules of M and investigate some properties of this class of graded modules.