Graded weakly 1-absorbing primary ideals

Abstract

Abstract Let G G be a group and R R be a G G -graded commutative ring with nonzero unity 1. In this article, we introduce the concept of graded weakly 1-absorbing primary ideals which is a generalization of graded 1-absorbing primary ideal. A proper graded ideal P P of R R is said to be a graded weakly 1-absorbing primary ideal of R R if whenever nonunit elements x , y , z ∈ h ( R ) x,y,z\in h\left(R) such that 0 ≠ x y z ∈ P 0\ne xyz\in P , then x y ∈ P xy\in P or z n ∈ P {z}^{n}\in P , for some n ∈ N n\in {\mathbb{N}} . Several properties of graded weakly 1-absorbing primary ideals are investigated.