Commutative weakly invo-clean group rings. (English) Zbl 1465.16020

A ring \(R\) is said to be invo-clean if, for every \(r\in R\), there exist an involution \(v\) and an idempotent \(e\) such that \(r=v+e\). If \(r=v+e\) or \(r=v-e\), the ring is called weakly invo-clean.
The following result is proved as
Theorem 1. Let \(G\) be an abelian group and let \(R\) be a commutative non-zero ring. Then the group ring \(R[G]\) is weakly invo-clean if and only if at most one of the following is true:
(1) \(G=\{1\}\) and \(R\) is weakly invo-clean.
(2) \(G\neq \{1\}\) and \(R\cong R_{1}\times R_{2}\) is invo-clean such that either
(2.1) \(\left\vert G\right\vert >2\), \(G^{2}=\{1\}\), \(R_{1}=\{0\}\) or \(R_{1}\) is a ring of \(\mathrm{char}(R_{1})=2\), and \(R_{2}=\{0\}\) or \(R_{2}\) is a ring of \(\mathrm{char}(R_{2})=3\)
(2.2) \(\left\vert G\right\vert =2\), \(2r_{1}^{2}=2r_{1}\) for all \(r_{1}\in R_{1}\) (in addition \(4=0\) in \(R_{1}\)), and \(R_{2}=\{0\}\) or \(R_{2}\) is a ring of \(\mathrm{char}(R_{2})=3\).


16S34 Group rings
16U99 Conditions on elements
Full Text: DOI MNR


[1] Danchev P. V., “Invo-clean unital rings”, Commun. Korean Math. Soc., 32:1 (2017), 19-27 · Zbl 1357.16054
[2] Danchev P. V., “Weakly invo-clean unital rings”, Afr. Mat., 28:7-8 (2017), 1285-1295 · Zbl 1380.16026
[3] Danchev P. V., “Feebly invo-clean unital rings”, Ann. Univ. Sci. Budapest (Math.), 60 (2017), 85-91 · Zbl 1426.16034
[4] Danchev P. V., “Weakly semi-boolean unital rings”, JP J. Algebra Number Theory Appl., 39:3 (2017), 261-276 · Zbl 1373.16066
[5] Danchev P. V., “Commutative invo-clean group rings”, Univ. J. Math. Math. Sci., 11:1 (2018), 1-6 · Zbl 1426.16021
[6] Danchev P. V., McGovern W. Wm., “Commutative weakly nil clean unital rings”, J. Algebra, 425:5 (2015), 410-422 · Zbl 1316.16028
[7] Karpilovsky G., “The Jacobson radical of commutative group rings”, Arch. Math. (Basel), 39:5 (1982), 428-430 · Zbl 0487.16012
[8] McGovern W. Wm., Raja Sh., Sharp A., “Commutative nil clean group rings”, J. Algebra Appl., 14:6 (2015), art. no. 1550094 · Zbl 1325.16024
[9] Milies C. P., Sehgal S. K., An Introduction to Group Rings, Springer, Netherlands, 2002, 371 pp. · Zbl 0997.20003
[10] Passman D. S., The Algebraic Structure of Group Rings, Dover Publications, New York, 2011, 752 pp.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.