Strongly P-Clean and Semi-Boolean Group Rings

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.

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Correspondence to D. Udar or R. K. Sharma or J. B. Srivastava.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 12, pp. 1717–1722, December, 2019.

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Udar, D., Sharma, R.K. & Srivastava, J.B. Strongly P-Clean and Semi-Boolean Group Rings. Ukr Math J 71, 1965–1971 (2020).

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