On Graded ϕ-1-absorbing prime ideals
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Let G be a group, R be a G-graded commutative ring with nonzero unity and GI(R) be the set of all graded ideals of R. Suppose that φ : GI(R) → GI(R) ∪ {∅} is a function. In this article, we introduce and study the concept of graded φ-1-absorbing prime ideals. A proper graded ideal I of R is called a graded φ-1-absorbing prime ideal of R if whenever a, b, c are homogeneous nonunit elements of R such that abc ∈ I − φ(I), then ab ∈ I or c ∈ I. Several properties of graded φ-1-absorbing prime ideals have been examined.
Abstract
Let G be a group, R be a G-graded commutative ring with nonzero unity and GI(R) be the set of all graded ideals of R. Suppose that φ : GI(R) → GI(R) ∪ {∅} is a function. In this article, we introduce and study the concept of graded φ-1-absorbing prime ideals. A proper graded ideal I of R is called a graded φ-1-absorbing prime ideal of R if whenever a, b, c are homogeneous nonunit elements of R such that abc ∈ I − φ(I), then ab ∈ I or c ∈ I. Several properties of graded φ-1-absorbing prime ideals have been examined.
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