On n-semiprimary ideals and n-pseudo valuation domains

Abstract

Let R be a commutative ring with $1\cancel{=}0$ and n a positive integer. A proper ideal I of R is an n-semiprimary ideal of R if whenever $x^n{y}^n\in I$ for $x,y\in R,$ then $x^n\in I$ or $y^n\in I.$ Let R be an integral domain with quotient field K. A proper ideal I of R is an n-powerful ideal of R if whenever $x^n{y}^n\in I$ for $x,y\in K,$ then $x^n\in R$ or $y^n\in R;$ and I is an n-powerful semiprimary ideal of R if whenever $x^n{y}^n\in I$ for $x,y\in K,$ then $x^n\in I$ or $y^n\in I.$ If every prime ideal of R is an n-powerful semiprimary ideal of R, then R is an n-pseudo-valuation domain (n-PVD). In this paper, we study the above concepts and relate them to several generalizations of pseudo-valuation domains.

Keywords:

• 2-absorbing ideal
• n-absorbing ideal
• powerful ideal
• primary ideal
• prime ideal
• pseudo-almost valuation domain
• pseudo-valuation domain
• radical ideal
• semiprimary ideal
• valuation domain

MATHEMATICS SUBJECT CLASSIFICATION (2000):

• 13A15

Acknowledgment

We would like to thank the referee for a careful reading of the paper.