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RINGS WITH A FINITE NUMBER OF ORBITS UNDER THE REGULAR ACTION
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 Title & Authors
RINGS WITH A FINITE NUMBER OF ORBITS UNDER THE REGULAR ACTION
Han, Juncheol; Park, Sangwon;
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 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 , , if a, b are singular matrices of the same rank, then , where and 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 , , with infinite division rings of the same cardinalities or R is isomorphic to the ring of matrices over a finite field if and only if for all .
 Keywords
left (right) regular action;orbit;left Artinian ring;
 Language
English
 Cited by
1.
Structure of Abelian rings, Frontiers of Mathematics in China, 2016  crossref(new windwow)
 References
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2.
J. Han, Regular action in a ring with a finite number of orbits, Comm. Algebra 25 (1997), no. 7, 2227-2236. crossref(new window)

3.
Y. Hirano, Rings with finitely many orbits under the regular action, Rings, modules, algebras, and abelian groups, 343-347, Lecture Notes in Pure and Appl. Math., 236, Dekker, New York, 2004.

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W. K. Nicholson, Introduction to Abstract Algebra, PWS, Boston, 1998.