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.