2021/04/19 by Alahmadi, A., Jain, S. K., Leroy, A.
#16E50 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2104.09296
Separative von Neumann regular rings exist in abundance. For example, all regular self-injective rings, unit regular rings, regular rings with a polynomial identity are separative. It remains open whether there exists a non-separative regular ring. In this note, we study a variety of conditions under which a von Neumann regular ring is separative. We show that a von Neumann regular ring R is separative under anyone of the following cases: \it (1) R is CS; \it (2) R is pseudo injective (auto-injective); \it (3) R satisfies the closure extension property: the essential closures in R of two isomorphic right ideals are themselves isomorphic. We also give another characterization of a regular perspective ring (Proposition 3.3)