Regularity of Idempotent Reflexive GP-V’-Rings

This paper discusses the regularity of the GP-V’-rings in conjunction with idempotent reflexivity for the first time. We mainly discuss the weak and strong regularity of the GP-V’-rings using generalized weak ideals, weakly right ideals, and quasi-ideals. We show the following: (1) If R is an idempotent reflexive semi-abelian left GP-V’-ring whose every maximal essential left ideal is a generalized weak ideal, a weakly right ideal, or a quasi-ideal, then R is a reduced left weakly regular ring. (2) R is a strongly regular ring if and only if R is an idempotent reflexive semi-commutative left GP-V’-ring whose every maximal essential left ideal is a generalized weak ideal, a weakly right ideal, or a quasi-ideal. (3) If R is a semi-primitive idempotent reflexive ring whose every simple singular left R-module is flat, and every maximal left ideal is a generalized weak ideal, then, for any nonzero element a∈R, there exists a positive integer n such that an≠0, and RaR+lan=R.