On Graded $ϕ$-$1$-absorbing prime ideals

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.

On graded pseudo 2-prime ideals

“…Recall from that [23], a G-graded ring R, where G is a group, is said to be a graded von Neumann regular ring if for each a ∈ R g (g ∈ G), there exists x ∈ R g −1 such that a = a 2 x. Also, a graded commutative ring R with unity is said to be a graded field if every nonzero homogeneous element of R is unit [26].…”

“…Clearly, every field is a graded field, however, the converse is not necessarily true, see ( [26], Example 3.6). By ( [23], Example 3.3), every graded field is a graded von Neumann regular ring. In the same context, a graded commutative ring R with unity is said to be a graded domain if R has no homogeneous zero divisor.…”

“…2 By ( [23], Lemma 3.7), if R is a graded von Neumann regular ring, where G is a group and x ∈ h(R), then Rx = Ra for some idempotent element a ∈ R e . So, (Rx) n = Rx for each positive integer n, and then Grad(Rx) = Rx.…”