$V$-rings versus $\Sigma$-$V$ Rings

TL;DR

This paper explores the properties of $V$-rings and $ ext{Sigma}$-$V$ rings, establishing new analogues of classical results and revealing that exchange $ ext{Sigma}$-$V$ rings are von Neumann regular and symmetric.

Contribution

It introduces analogues of known $V$-ring results for $ ext{Sigma}$-$V$ rings and proves that exchange $ ext{Sigma}$-$V$ rings are von Neumann regular and symmetric.

Findings

Prime and primitive notions coincide in $ ext{Sigma}$-$V$ rings.

Exchange $ ext{Sigma}$-$V$ rings are von Neumann regular.

$ ext{Sigma}$-$V$ rings exhibit left-right symmetry.

Abstract

This paper studies similarities and differences between the classes of rings over which each simple module is injective and rings over which each simple module is $\Sigma$-injective. The rings in the former class are called $V$-rings and the rings in the latter class are called $\Sigma$-$V$ rings. We have obtained analogues of various well-known results about $V$-rings for $\Sigma$-$V$ rings. Motivated by a conjecture of Kaplansky, Fisher asked if a prime right $V$-ring is right primitive. Although a counter-example to Kaplansky's conjecture was constructed long ago but Fisher's question is still open. In this paper we show that for a right $\Sigma$-$V$ ring, the notions of prime and primitive are equivalent. Also, we show that an exchange $\Sigma$-$V$ ring is left-right symmetric and moreover, it is von Neumann regular.