Nil-clean group rings

Abstract

An element a of a ring R is nil-clean, if a = e + b, where e(2) = e is an element of R and b is a nilpotent element, and the ring R is called nil-clean if each of its elements is nil-clean. In [W. Wm. McGovern, S. Raja and A. Sharp, Commutative nil clean group rings, J. Algebra Appl. 14(6) (2015) 5; Article ID: 1550094], it was proved that, for a commutative ring R and an abelian group G, the group ring RG is nil-clean, iff R is nil-clean and G is a 2-group. Here, we discuss the nil-cleanness of group rings in general situation. We prove that the group ring of a locally finite 2-group over a nil-clean ring is nil-clean, and that the hypercenter of the group G must be a 2-group if a group ring of G is nil-clean. Consequently, the group ring of a nilpotent group over an arbitrary ring is nil-clean, iff the ring is a nil-clean ring and the group is a 2-group.