Structure of NI rings related to centers

Abstract

We first obtain that NI rings satisfy a property that if ab is central for elements a, b, then for some , by applying a property of reduced rings. We prove next the following: Let R be a ring and I be the ideal of R generated by the subset . (i) Suppose that ab is central for and ab – ba is a nonzero nilpotent. Then, is a nonzero nilpotent ideal of the subring A of R, where 1 is the identity of R, , and A is the algebra generated by a, b over B. (ii) If R is NI, then I is nil and R/I is an Abelian NI ring. (iii) Let R be reversible and ab be central for . Then, there exists such that, for every and for all ; especially . We call a ring pseudo-NI if it satisfies the first property of NI rings to be mentioned and examine the structures of NI and pseudo-NI rings in several ring theoretic situations, showing that semisimple Artinian rings are pseudo-NI.