2016/05/17 by Bijan Davvaz, Zahra Nazemian, Davvaz, Bijan +3
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Rings, Modules, and Algebras #math.RA #msc:13E99 #msc:16P99
paper · pdf · doi:10.48550/arxiv.1605.05009
arxiv created 2016/05/17 · arxiv updated 2016/05/18
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.