RINGS WITH A FINITE NUMBER OF ORBITS UNDER THE REGULAR ACTION

Title & Authors
RINGS WITH A FINITE NUMBER OF ORBITS UNDER THE REGULAR ACTION
Han, Juncheol; Park, Sangwon;

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 $\small{M_n(D)}$, $\small{n{\geq}2}$, if a, b are singular matrices of the same rank, then $\small{{\mid}o_{\ell}(a){\mid}={\mid}o_{\ell}(b){\mid}}$, where $\small{o_{\ell}(a)}$ and $\small{o_{\ell}(b)}$ 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 $\small{X(R){\neq}{\emptyset}}$, $\small{R{{\sim_=}}{\oplus}^m_{i=1}M_n_i(D_i)}$, with $\small{D_i}$ infinite division rings of the same cardinalities or R is isomorphic to the ring of $\small{2{\times}2}$ matrices over a finite field if and only if $\small{{\mid}o_{\ell}(x){\mid}={\mid}o_{\ell}(y){\mid}}$ for all $\small{x,y{\in}X(R)}$.
Keywords
left (right) regular action;orbit;left Artinian ring;
Language
English
Cited by
1.
Structure of Abelian rings, Frontiers of Mathematics in China, 2017, 12, 1, 117
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