# GROUP ACTIONS IN A REGULAR RING

• HAN, Jun-Cheol (DEPARTMENT OF MATHEMATICS EDUCATION, PUSAN NATIONAL UNIVERSITY)
• Published : 2005.11.01

#### 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 $\in$ X. Secondly, if F is a field in which 2 is a unit and F $\backslash\;\{0\}$ 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.

#### 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 https://doi.org/10.1080/00927879708825984
4. J. Han, Group actions in a unit-regular ring, Comm. Algebra 27 (1999), no. 7, 3353-3361 https://doi.org/10.1080/00927879908826632

#### Cited by

1. Group Action on Fuzzy Modules vol.07, pp.05, 2016, https://doi.org/10.4236/am.2016.75038