On clean and semi-clean group rings

A ring [Formula: see text] is called clean (respectively, semi-clean) if each element of [Formula: see text] can be written as the sum of a unit and an idempotent (respectively, a periodic element). In this paper, we seek the necessary and sufficient conditions under which, for a ring [Formula: see text] and a group [Formula: see text] with additional conditions, the group ring [Formula: see text] is clean or semi-clean. As a remarkable result, let [Formula: see text] be an Abelian clean ring, a reduced ring, or a commutative ring, and let [Formula: see text] be a locally nilpotent group, we show that, if [Formula: see text] is semi-clean, then [Formula: see text] is locally finite. Also, we show that if [Formula: see text] is a semi-local ring whose Jacobson radical is locally nilpotent and [Formula: see text] is a locally finite group, then [Formula: see text] is clean. These results generalize some earlier results in the literature.