Nil-clean and strongly nil-clean rings

Abstract

An element a of a ring R is nil-clean if a=e+b where e2=e∈R and b is a nilpotent; if further eb=be, the element a is called strongly nil-clean. The ring R is called nil-clean (resp., strongly nil-clean) if each of its elements is nil-clean (resp., strongly nil-clean). It is proved that an element a is strongly nil-clean iff a is a sum of an idempotent and a unit that commute and a−a2 is a nilpotent, and that a ring R is strongly nil-clean iff R/J(R) is boolean and J(R) is nil, where J(R) denotes the Jacobson radical of R. The strong nil-cleanness of Morita contexts, formal matrix rings and group rings is discussed in details. A necessary and sufficient condition is obtained for an ideal I of R to have the property that R/I strongly nil-clean implies R is strongly nil-clean. Finally, responding to the question of when a matrix ring is nil-clean, we prove that the matrix ring over a 2-primal ring R is nil-clean iff R/J(R) is boolean and J(R) is nil, i.e., R is strongly nil-clean.