Properties of ϕ-Primal Graded Ideals

• Main Author: Ameer Jaber
• Format: Article
• Language: English
• Published: Hindawi Limited 2017-01-01
• Series: Journal of Mathematics
• Online Access: http://dx.doi.org/10.1155/2017/3817479

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.