Properties of ϕ-Primal Graded Ideals

Abstract EN

Let R be a commutative graded ring with unity 1≠0.

A proper graded ideal of R is a graded ideal I of R such that I≠R.

Let ϕ:I(R)→I(R)∪{∅} be any function, where I(R) denotes the set of all proper graded ideals of R.

A homogeneous element a∈R is ϕ-prime to I if ra∈I-ϕ(I) where r is a homogeneous element in R; then r∈I.

An element a=∑g∈Gag∈R is ϕ-prime to I if at least one component ag of a is ϕ-prime to I.

Therefore, a=∑g∈Gag∈R is not ϕ-prime to I if each component ag of a is not ϕ-prime to I.

We denote by νϕ(I) the set of all elements in R that are not ϕ-prime to I.

We define I to be ϕ-primal if the set P=νϕ(I)+ϕ(I) (if ϕ≠ϕ∅) or P=νϕ(I) (if ϕ=ϕ∅) forms a graded ideal of R.

In the work by Jaber, 2016, the author studied the generalization of primal superideals over a commutative super-ring R with unity.

In this paper we generalize the work by Jaber, 2016, to the graded case and we study more properties about this generalization.