Nil-clean group rings

Abstract

An element [Formula: see text] of a ring [Formula: see text] is nil-clean, if [Formula: see text], where [Formula: see text] and [Formula: see text] is a nilpotent element, and the ring [Formula: see text] 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 [Formula: see text] and an abelian group [Formula: see text], the group ring [Formula: see text] is nil-clean, iff [Formula: see text] is nil-clean and [Formula: see text] is a [Formula: see text]-group. Here, we discuss the nil-cleanness of group rings in general situation. We prove that the group ring of a locally finite [Formula: see text]-group over a nil-clean ring is nil-clean, and that the hypercenter of the group [Formula: see text] must be a [Formula: see text]-group if a group ring of [Formula: see text] 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 [Formula: see text]-group.