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GROUP ACTIONS IN A REGULAR RING
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 Title & Authors
GROUP ACTIONS IN A REGULAR RING
HAN, Jun-Cheol;
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 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 X. Secondly, if F is a field in which 2 is a unit and F 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  crossref(new windwow)
 References
1.
K. R. Goodearl, von Neumann Regualr Rings, Pitman Publishing Limited, London, 1979

2.
J. Han, The group of units in a left Artinian ring, Bull. Korean Math. Soc. 31 (1994), no. 1, 99-104

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

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