# RINGS WITH A FINITE NUMBER OF ORBITS UNDER THE REGULAR ACTION

• Han, Juncheol (Department of Mathematics Education Pusan National University) ;
• Park, Sangwon (Department of Mathematics Dong-A University)
• Received : 2012.07.02
• Published : 2014.07.01
• 63 5

#### Abstract

Let R be a ring with identity, X(R) the set of all nonzero, non-units of R and G(R) the group of all units of R. We show that for a matrix ring $M_n(D)$, $n{\geq}2$, if a, b are singular matrices of the same rank, then ${\mid}o_{\ell}(a){\mid}={\mid}o_{\ell}(b){\mid}$, where $o_{\ell}(a)$ and $o_{\ell}(b)$ are the orbits of a and b, respectively, under the left regular action. We also show that for a semisimple Artinian ring R such that $X(R){\neq}{\emptyset}$, $$R{{\sim_=}}{\oplus}^m_{i=1}M_n_i(D_i)$$, with $D_i$ infinite division rings of the same cardinalities or R is isomorphic to the ring of $2{\times}2$ matrices over a finite field if and only if ${\mid}o_{\ell}(x){\mid}={\mid}o_{\ell}(y){\mid}$ for all $x,y{\in}X(R)$.

#### Keywords

left (right) regular action;orbit;left Artinian ring

#### Acknowledgement

Supported by : Dong-A University

#### References

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#### Cited by

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2. Structure of Abelian rings vol.12, pp.1, 2017, https://doi.org/10.1007/s11464-016-0586-z