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 ( R n ) = ( M ( R)) n , where R n 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 a 2 x = 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.
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Tetsuo Kandô (1952) studied this question.