Uniquely weakly nil-clean conditions on zero-divisors
An element in a ring $R$ is called uniquely weakly nil-clean if every element in $R$ can be uniquely written as a sum or a difference of a nilpotent and an idempotent in the sense of very idempotents. The structure of the ring 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 $F$, ${\Bbb Z}_{3}\oplus {\Bbb Z}_{3}$, ${\Bbb Z}_{3}\oplus B$ 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 also considered. Furthermore, we prove that every zero-divisor in a ring $R$ is uniquely nil-clean if and only if $R$ is a D-ring, or $R$ is abelian, periodic; and $R/J(R)$ is Boolean.\vskip3mm \no {\bf Key words}: Zero-divisor; Uniquely weakly nil-clean ring; Uniquely nil-clean ring.