Structure of rings with a condition on zero divisors (Q2758077)

For an associative ring \(R\) we denote \(J(R)\) the Jacobson radical of \(R\), \(C(R)\) the commutator of \(R\) and \(N(R)\) the set of nilpotent elements of \(R\). The purpose of this paper is the study of \(D\)-rings. A \(D\)-ring is a ring in which every (left or right) zero divisor is nilpotent.NEWLINENEWLINENEWLINESome of results of this paper are presented in the following Theorem. Let \(R\) be a \(D\)-ring. (i) If \(N\) is an ideal of \(R\) then \(R=N\) or \(R/N\) is a domain. (ii) If \(R\) is weakly periodic then \(R\) is periodic, \(C(R)\) is nil and \(R\) is either nil or local. (iii) If \(R\) is Artinian and \(R\neq N\) then \(N=J\), \(R/N\) is a domain, \(R\) is local and has an identity. (iv) If \(R\) is semiperfect and \(R\neq J\) then \(R\) is local. (v) If \(R\) is semiprime, \(R\neq N\) and \(R\) satisfies a polynomial identity then \(R\) is a domain.