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Research articles
Commutative weakly tripotent group rings
Peter V. Danchev Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, "Acad. G. Bonchev", str., bl. 8, 1113 Sofia, Bulgaria
Аннотация:
Very recently, Breaz and Cîmpean introduced and examined in Bull. Korean Math. Soc. (2018) the class of so-called
weakly tripotent rings as those rings
$R$ whose elements satisfy at leat one of the equations
$x^3=x$ or
$(1-x)^3=1-x$. These rings are generally non-commutative. We here obtain a criterion when the commutative group ring
$RG$ is weakly tripotent in terms only of a ring
$R$ and of a group
$G$ plus their sections.
Actually, we also show that these weakly tripotent rings are
strongly invo-clean rings in the sense of Danchev in Commun. Korean Math. Soc. (2017). Thereby, our established criterion somewhat strengthens previous results on commutative strongly invo-clean group rings, proved by the present author in Univ. J. Math. & Math. Sci. (2018). Moreover, this criterion helps us to construct a commutative strongly invo-clean ring of characteristic
$2$ which is
not weakly tripotent, thus showing that these two ring classes are different.
Ключевые слова и фразы:
tripotent rings, weakly tripotent rings, strongly invo-clean rings, group rings.
MSC: 16S34,
16U99,
20C07 Поступила в редакцию: 18.11.2019
Язык публикации: английский