Graded $\delta$-primary structures

Abstract

Let $G$ be a group with identity $e$, $R$ a $G$-graded commutative ring with unity $1$ and $M$ a $G$-graded $R$-module. In this article, we unify the concepts of graded prime ideals and graded primary ideals into a new concept, namely, graded $\delta$-primary ideals. Also, we unify the concepts of graded $2$-absorbing ideals and graded $2$-absorbing primary ideals into a new concept, namely, graded $2$-absorbing $\delta$-primary ideals. A number of results about graded prime, graded primary, graded $2$-absorbing and graded $2$-absorbing primary ideals are extended into these new structures. Finally, we extend the concept of graded $\delta$-primary ideals into graded $\delta$-primary submodules. A number of results about graded prime, graded primary submodules are extended into this new structure.

On graded 1 -absorbing δ -primary ideals

“…A proper graded ideal I of R is said to be a graded δ-semiprimary ideal of R if whenever ab ∈ I for some nonunit homogeneous elements a, b ∈ I, then a ∈ δ(I) or b ∈ δ(I). Recall from [3] that a graded ideal is said to be graded 2-absorbing δ-primary ideal if whenever a, b, c ∈ h(R) and abc ∈ I, then ab ∈ I or ac ∈ δ(I) or bc ∈ δ(I).…”

“…[2, Theorem 1]. Let I(R) be the set of all ideals of a graded ring R. The authors in [3] introduced the concept of graded ideal expansion of a graded ring R. We recall from [3] that a function δ : I(R) −→ I(R) is called a graded ideal expansion of R if it assigns to every graded ideal I another graded ideal On graded 1-absorbing δ-primary ideals 573 δ(I) of R and if whenever L, I, J are graded ideals of R with J ⊆ I, we have L ⊆ δ(L) and δ(J) ⊆ δ(I). In addition, recall from [3] that a proper graded ideal I of R is said to be a graded δ-primary ideal of R if whenever a, b ∈ h(R) with ab ∈ I, we have a ∈ I or b ∈ δ(I), where δ is a graded ideal expansion of R.…”