On n-δ-semiprimary Ideals of Commutative Rings

Abstract

Let R be a commutative ring with identity and n a positive integer. A generalization of prime ideals is introduced in (Anderson and Badawi, 2021). A proper ideal J of R is said to be an n-semiprimary ideal if whenever a,b∈ R with a^n b^n∈ J, then a^n∈ J or b^n ∈J. Let δ:Id(R)⟶ Id(R) be an expansion function of ideals of R where Id(R) is the set of all ideals of R. The aim of this paper is to introduce the class of n-δ-semiprimary ideals generalizing the notion of n-semiprimary ideals. We call a proper ideal J of R an n-δ-semiprimary ideal if whenever a^n b^n∈ J for a,b∈ R, then a^n∈δ(J) or b^n∈δ(J). Several properties and characterizations regarding this class of ideals with many supporting examples are presented. Additionally, we call a proper ideal J of R a strongly n-δ-semiprimary ideal of R if whenever K^n L^n⊆ J for proper ideals K and L of R, then K^n⊆δ(J) or L^n⊆δ(J). We investigate the relationship between these two concepts. Moreover, the behaviour of n-δ-semiprimary ideals under homomorphisms, in localization rings, in division rings, in cartesian product of rings and in idealization rings is investigated.

Keywords

• prime ideals
• delta semiprimary ideals
• delta-primary ideal
• n-semiprimary ideal

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