Regularity of n-P-V-Rings and n-P-V’-Rings - Liuwen Li; Wenlin Zou; Ying Li (Axioms)
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 <semantics>π</semantics>-regularity and weak <semantics>π</semantics>-regularity of the n-P-V-rings and the n-P-V’-rings. Subsequently, we demonstrate our results as follows: (1) <semantics>R</semantics> is strongly <semantics>π</semantics>-regular if <semantics>R</semantics> 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) <semantics>R</semantics> is strongly <semantics>π</semantics>-regular iff <semantics>R</semantics> 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) <semantics>R</semantics> is reduced left weakly <semantics>π</semantics>-regular if <semantics>R</semantics> 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.