RINGS IN WHICH EVERY ZERO DIVISOR IS THE SUM OR DIFFERENCE OF A NILPOTENT ELEMENT AND AN IDEMPOTENT

An element in a ring R is called uniquely weakly nil-clean if it can be uniquely written as the sum or difference of a nilpotent element and an idempotent. The structure of rings in which every zero-divisor is uniquely weakly nil-clean is completely determined. We prove that every zero-divisor in a ring R is uniquely weakly nil-clean if and only if R is a D-ring, or R is abelian, periodic, and R/ J(R) is isomorphic to a field F, Z(3) circle plus Z(3), Z(3) circle plus B where B is Boolean, or a Boolean ring. As a specific case, rings in which every zero-divisor a or -a is a nilpotent or an idempotent are characterized. Furthermore, we prove that every zero-divisor in a ring R can be uniquely written as the sum of a nilpotent element and an idempotent if and only if R is a D-ring, or R is abelian, periodic and R/ J(R) is Boolean.