Strongly unit nil-clean rings

摘要

An element in a ring is strongly nil-clean, if it is the sum of an idempotent and a nilpotent element that commute. A ring R is strongly unit nil-clean, if for any a is an element of R there exists a unit u is an element of R, such that ua is strongly nil-clean. We prove, in this paper, that a ring R is strongly unit nil-clean, if and only if every element in R is equivalent to a strongly nil-clean element, if and only if for any a is an element of R, there exists a unit ua is an element of R, such that ua. R is strongly pi-regular. Strongly unit nil-clean matrix rings are investigated as well.