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 represented as the sum of an idempotent and a strongly nilpotent element, which are commuting. 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 are lifted 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 either a locally finite soluble group or an FC group.

中文翻译：

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强 P-Clean 和半布尔群环

如果每个元素都是（唯一表示为）幂等和单元的总和，则环 R 被称为干净（相应地，唯一干净）。如果一个环 R 的每个元素都可以表示为一个幂等元素和一个强幂零元素之和，那么它们被称为强 P-clean。强 P 清洁环类是半布尔清洁环和强零清洁环类的子类。如果 R/J ( R ) 是布尔值且幂等项以 J ( R ) 为模被提升，则环 R 被称为半布尔环，其中 J ( R ) 表示 R 的雅各布森根。半布尔环的类别严格介于类独特的干净和干净的戒指。我们获得了强 P-clean 群环的完整表征。证明了群环RG 是强P-clean 当且仅当R 是强P-clean 且G 是局部有限2-群。更多，我们还研究了半布尔群环。证明了如果群环 RG 是半布尔环，则 R 是半布尔环，G 是 2-群，如果 G 是局部有限可解群或 FC 群，则相反的断言为真。