Regularity of n-P-V-Rings and n-P-V’-Rings

Abstract

The regularity of the n-P-V-rings and n-P-V’-rings is systematically investigated in this paper. Employing the notions of quasi-ideals, weakly left (or right) ideals, and generalized weak ideals, we focus on investigating the strong -regularity and weak -regularity of the n-P-V-rings and the n-P-V’-rings. Subsequently, we demonstrate our results as follows: (1) is strongly -regular if is a left n-P-V-ring where all its maximal left ideals are either quasi-ideals, weakly right ideals, or generalized weak ideals. (2) is strongly -regular iff is an abelian left (right) n-P-V’-ring where all its maximal essential left (right) ideals are either quasi-ideals, weakly right (left) ideals, or generalized weak ideals. (3) is a reduced left weakly -regular if is an idempotent reflexive semi-abelian left n-P-V’-ring where all its maximal essential left ideals are either quasi-ideals, weakly right ideals, or generalized weak ideals.