Regularity of Idempotent Reflexive GP-V’-Rings

Liuwen Li, Wenlin Zou () and Ying Li
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Liuwen Li: School of Science, Jinling Institute of Technology, Nanjing 211169, China
Wenlin Zou: School of Information Engineering, Nanjing Xiaozhuang University, Nanjing 211171, China
Ying Li: School of Information Engineering, Nanjing Xiaozhuang University, Nanjing 211171, China

Mathematics, 2024, vol. 12, issue 20, 1-11

Abstract: 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 a n ≠ 0 , and R a R + l a n = R .

Keywords: regularity; GP-V’-rings; semi-primitive rings; abelian rings (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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