$V$-rings versus $Σ$-$V$ Rings

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 $Σ$-injective. The rings in the former class are called $V$-rings and the rings in the latter class are called $Σ$-$V$ rings. We have obtained analogues of various well-known results about $V$-rings for $Σ$-$V$ 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 $Σ$-$V$ ring, the notions of prime and primitive are equivalent. Also, we show that an exchange $Σ$-$V$ ring is left-right symmetric and moreover, it is von Neumann regular.

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