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Volume 28, Issue 1
Uniquely Strongly Clean Group Rings

Xiulan Wang

Commun. Math. Res., 28 (2012), pp. 17-25.

Published online: 2021-05

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  • Abstract

A ring $R$ is called clean if every element is the sum of an idempotent and a unit, and $R$ is called uniquely strongly clean (USC for short) if every element is uniquely the sum of an idempotent and a unit that commute. In this article, some conditions on a ring $R$ and a group $G$ such that $RG$ is clean are given. It is also shown that if $G$ is a locally finite group, then the group ring $RG$ is USC if and only if $R$ is USC, and $G$ is a 2-group. The left uniquely exchange group ring, as a middle ring of the uniquely clean ring and the USC ring, does not possess this property, and so does the uniquely exchange group ring.

  • Keywords

clean ring, group ring, $p$-group, USC ring.

  • AMS Subject Headings

16S34, 16N40

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-28-17, author = {Wang , Xiulan}, title = {Uniquely Strongly Clean Group Rings}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {28}, number = {1}, pages = {17--25}, abstract = {

A ring $R$ is called clean if every element is the sum of an idempotent and a unit, and $R$ is called uniquely strongly clean (USC for short) if every element is uniquely the sum of an idempotent and a unit that commute. In this article, some conditions on a ring $R$ and a group $G$ such that $RG$ is clean are given. It is also shown that if $G$ is a locally finite group, then the group ring $RG$ is USC if and only if $R$ is USC, and $G$ is a 2-group. The left uniquely exchange group ring, as a middle ring of the uniquely clean ring and the USC ring, does not possess this property, and so does the uniquely exchange group ring.

}, issn = {2707-8523}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cmr/19061.html} }
TY - JOUR T1 - Uniquely Strongly Clean Group Rings AU - Wang , Xiulan JO - Communications in Mathematical Research VL - 1 SP - 17 EP - 25 PY - 2021 DA - 2021/05 SN - 28 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cmr/19061.html KW - clean ring, group ring, $p$-group, USC ring. AB -

A ring $R$ is called clean if every element is the sum of an idempotent and a unit, and $R$ is called uniquely strongly clean (USC for short) if every element is uniquely the sum of an idempotent and a unit that commute. In this article, some conditions on a ring $R$ and a group $G$ such that $RG$ is clean are given. It is also shown that if $G$ is a locally finite group, then the group ring $RG$ is USC if and only if $R$ is USC, and $G$ is a 2-group. The left uniquely exchange group ring, as a middle ring of the uniquely clean ring and the USC ring, does not possess this property, and so does the uniquely exchange group ring.

Wang , Xiulan. (2021). Uniquely Strongly Clean Group Rings. Communications in Mathematical Research . 28 (1). 17-25. doi:
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