Abstract
Let Rbe a unit-regular ring , let Xbe the set of all nonzero, nonunits of Rand let Gbe the group of all units of R. In this paper, some finiteness properties of Rare investigated by considering group actions of Gon Xas follows:First, in case of half-transitive regualr action if 2 is unit in Ror the number of idempotents in Ris finite, then Ris finite. Secondly, if Gis cyclic and 2 is unit in R, then every orbit under regualr action is a finite set, and so in this case, if Rhas a finite number of idempotents, then Ris finite. Finally, if Fis a field in which 2 is unit and the multiplicative group of all nonzero elenents in Fforms a cyclic group, then Fis finite.
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