On graded 1 -absorbing δ -primary ideals

Abstract

Let G be an abelian group with identity 0 and let R be a commutative graded ring of type G with nonzero unity. Let I(R) be the set of all ideals of R and let δ: I(R)⟶I(R) be a function. Then, according to (R. Abu-Dawwas, M. Refai, Graded δ-Primary Structures, Bol. Soc. Paran. Mat., 40 (2022), 1-11), δ is called a graded ideal expansion of a graded ring R if it assigns to every graded ideal I of R another graded ideal δ(I) of R with I ⊆ δ(I), and if whenever I and J are graded ideals of R with J ⊆ I, we have δ (J) ⊆ δ(I). Let δ be a graded ideal expansion of a graded ring R. In this paper, we introduce and investigate a new class of graded ideals that is closely related to the class of graded δ-primary ideals. A proper graded ideal I of R is said to be a graded 1-absorbing δ-primary ideal if whenever nonunit homogeneous elements a,b,c ∊ R with abc ∊ I, then ab ∊  I or c ∊  δ(I). After giving some basic properties of this new class of graded ideals, we generalize a number of results about 1-absorbing δ-primary ideals into these new graded structure. Finally, we study the graded 1-absorbing δ-primary ideals of the localization of graded rings and of the trivial graded ring extensions.