Structure of rings with certain conditions on zero divisors

Throughout this paper, R is an associative ring; andN ,C,C(R), and J denote, respectively, the set of nilpotent elements, the center, the commutator ideal, and the Jacobson radical. An element x of R is called potent if xn = x for some positive integer n= n(x) > 1. A ring R is called periodic if for every x in R, xm = xn for some distinct positive integersm=m(x), n = n(x). A ring R is called weakly periodic if every element of R is expressible as a sum of a nilpotent element and a potent element of R : R=N +P, where P is the set of potent elements of R. A ring R such that every zero divisor is nilpotent is called a D-ring. The structure of certain classes ofD-rings was studied in [1]. Following [7], R is called normal if all of its idempotents are in C. A ring R is called a D∗-ring, if every zero divisor x in R can be written as x = a+ b, where a ∈ N , b ∈ P, and ab = ba. Clearly every D-ring is a D∗-ring. In particular every nil ring is a D∗-ring, and every domain is a D∗-ring. A Boolean ring is a D∗-ring but not a D-ring. Our objective is to study the structure of certain classes of D∗-ring.