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arXiv · 2104.09296

Remarks on separativity of regular rings

A. Alahmadi·S. K. Jain·A. Leroy
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Abstract

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)

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A. Alahmadi, S. K. Jain, A. Leroy. 2021-04-19. Remarks on separativity of regular rings. https://arxiv.org/abs/2104.09296

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