Strongly $P$ -clean and semi-Boolean group rings

D. Udar; R. K. Sharma; J. B. Srivastava

D. Udar Indian Inst. Technology, Delhi, India
R. K. Sharma Indian Inst. Technology, Delhi, India
J. B. Srivastava Indian Inst. Technology, Delhi, India

Keywords: Strongly P-clean rings, semiboolean rings, group rings, clean 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.

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ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12

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Received 07.06.16