On strongly regular near-rings

Abstract

According to Mason [1] a right near-ring N is called (i) left (right) strongly regular if for every a there is an x in N such that a = xa2 (a = a2x) and (ii) left (right) regular if for every a there is an x in N such that a = xa2 (a = a2x) and a = axa. He proved that for a zerosymmetric near-ring with identity, the notions of left regularity, right regularity and left strong regularity are equivalent. The aim of this note is to prove that these three notions are equivalent for arbitrary near-rings. We also show that if N satisfies dec on iV-subgroups, then all the above four notions are equivalent.