Strongly P-Clean and Semi-Boolean Group Rings

Abstract

A ring $R$ is called clean (resp., uniquely clean) if every element is (uniquely represented as) the sum of an idempotent and a unit. A ring $R$ is called strongly P-clean if every its element can be written as the sum of an idempotent and a strongly nilpotent element that commute. The class of strongly P-clean rings is a subclass of classes of semi-Boolean and strongly nil clean rings. A ring $R$ is called semi-Boolean if $R/J(R)$ is Boolean and idempotents lift modulo $J(R),$ where $J(R)$ denotes the Jacobson radical of $R.$ The class of semi-Boolean rings lies strictly between the classes of uniquely clean and clean rings. We obtain a complete characterization of strongly P-clean group rings. It is proved that the group ring $RG$ is strongly P-clean if and only if $R$ is strongly P-clean and $G$ is a locally finite 2-group. Further, we also study semi-Boolean group rings. It is proved that if a group ring $RG$ is semi-Boolean, then $R$ is a semi-Boolean ring and $G$ is a 2-group and that the converse assertion is true if $G$ is locally finite and solvable, or an FC group.