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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 = {Xiulan and Wang and and 18438 and and Xiulan Wang}, 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.

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