Near-rings in which each element is a power of itself

Abstract

Let R denote a near-ring such that for each x ∈ R , there exists an integer n ( x ) > 1 for which x n(x) = x . We show that the additive group of R is commutative if 0.x; = 0 for all x ∈ R and every non-trivial homomorphic image R¯ of R contains a non-zero idempotent e commuting multiplicatively with all elements of R¯. As the major consequence, we obtain the result that if R is distributively-generated, then R is a ring – a generalization of a recent theorem of Ligh on boolean near-rings.