A structure theorem for finite Abelian regular rings

A regular (von Neumann regular) ring [Formula: see text] is said to be an Abelian regular ring if all the idempotents are central. In general, Abelian regular rings need not be commutative. In this paper, we discuss some properties of Abelian regular rings. Also, we describe the structure of Abelian regular rings in which the set of idempotents satisfying some minimality conditions and deduces that any finite Abelian regular ring is isomorphic to a direct product of fields and hence commutative.