A note on strongly clean elements

An element [Formula: see text] is (strongly) [Formula: see text]-clean provided that there exist an idempotent [Formula: see text] and an element [Formula: see text] such that [Formula: see text] (that commute, i.e. [Formula: see text]), where [Formula: see text] that is the largest Jacobson radical subring of [Formula: see text]. A ring [Formula: see text] is strongly [Formula: see text]-clean in case every element in [Formula: see text] is strongly [Formula: see text]-clean. It is shown that [Formula: see text] is a strongly [Formula: see text]-clean ring if and only if for every [Formula: see text], [Formula: see text] and each idempotent lifts strongly modulo [Formula: see text] (equivalently, [Formula: see text] is strongly clean). In this paper, we also describe the structure of group rings that are strongly [Formula: see text]-clean. It is shown that if [Formula: see text] is strongly [Formula: see text]-clean, then [Formula: see text] is strongly [Formula: see text]-clean where [Formula: see text]. In case [Formula: see text] is a locally finite [Formula: see text]-group in which every finite subgroup has an odd order, then [Formula: see text] is not strongly [Formula: see text]-clean.