Nil-clean group rings.

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