On the theory of pre-p-nil-rings

Abstract

Generalizing the concept of p-rings, Abian and Mcworter [1] call an associative and commutative ring R with characteristic p a pre-p-ring if xy^y = x^yy for every x and y in R. It was proved in [1] that every pre-p-ring R is a direct sum R = B ⊕ N of a p-ring B and a nil ring N, where even x^y+2 = 0 for every χ∈N . It was also proved in [1] that N is the radical of R and hence N uniquely determined by R. Moreover, it is not difficult to show that B is also uniquely determined by R. A simple calculation shows that the converse that the direct sum R = B ⊕ N of a p-ring B and a pre-p-nil-ring N is a pre-p-ring. Since the structure of p-rings is known, there remains to investigate only the pre-p-nil-rings, which is the purpose of this paper.


Journal of the Nigerian Association of Mathematical Physics Vol. 8 2004: pp. 273-276