GROUP ACTIONS IN A REGULAR RING

Title & Authors
GROUP ACTIONS IN A REGULAR RING
HAN, Jun-Cheol;

Abstract
Let R be a ring with identity, X the set of all nonzero, nonunits of Rand G the group of all units of R. We will consider two group actions on X by G, the regular action and the conjugate action. In this paper, by investigating two group actions we can have some results as follows: First, if G is a finitely generated abelian group, then the orbit O(x) under the regular action on X by G is finite for all nilpotents x $\small{\in}$ X. Secondly, if F is a field in which 2 is a unit and F $\small{\backslash\;\{0\}}$ is a finitley generated abelian group, then F is finite. Finally, if G in a unit-regular ring R is a torsion group and 2 is a unit in R, then the conjugate action on X by G is trivial if and only if G is abelian if and only if R is commutative.
Keywords
regular action;conjugate action;orbit;stablizer;transitive;bounded index;
Language
English
Cited by
1.
Group Action on Fuzzy Modules, Applied Mathematics, 2016, 07, 05, 413
References
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K. R. Goodearl, von Neumann Regualr Rings, Pitman Publishing Limited, London, 1979

2.
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3.
J. Han, Regular action in a ring with a finite number of orbits, Comm. Algebra 25 (1997), no. 7, 2227-2236

4.
J. Han, Group actions in a unit-regular ring, Comm. Algebra 27 (1999), no. 7, 3353-3361