Strong Regularity in Arbitrary Rings

Abstract

An element a of a ring R is called regular, if there exists an element x of R such that a×a = a, and a two-sided ideal a in R is said to be regular if each of its elements is regular B. Brown and N. H. McCoy [1] has recently proved that every ring R has a unique maximal regular two-sided ideal M(R), and that M(R) has the following radical-like property: (i) M(R/M(R)) = 0; (ii) if a is a two-sided ideal of R, then M(a) = a ∩ M(R); (iii) M(Rn) = (M(R))n, where Rn denotes a full matrix ring of order n over R. Arens and Kaplansky [2] has defined an element a of R to be strongly regular when there exists an element x of R such that a2x = a. We shall prove in this note that replacing “regularity” by “strong regularity,” we have also a unique maximal strongly regular ideal N(R), and shall investigate some of its properties.