An element a of a ring R is regular (in the sense of Von Neumann) if there exists x ϵ R such that axa = a. If every element of R is regular, then R is said to be a regular ring. 

(a) Every division ring is regular.

(b) A finite direct product of regular rings is regular.

(c) Every regular ring is semi simple. 

(d) The ring of all linear transformations on a vector space (not necessarily finite dimensional) over a division ring is regular.

(e) A semi simple left Artinian ring is regular.

(f) R is regular if and only if every principal left [resp. right] ideal of R is generated by an idempotent element.

(g) A nonzero regular ring R with identity is a division ring if and only if its only idempotents are 0 and 1_R.