DOI QR코드

DOI QR Code

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)
  • 투고 : 2012.07.02
  • 발행 : 2014.07.01

초록

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 $x,y{\in}X(R)$.

키워드

참고문헌

  1. J. Cohen and K. Koh, Half-transitive group actions in a compact ring, J. Pure Appl. Algebra 60 (1989), no. 2, 139-153. https://doi.org/10.1016/0022-4049(89)90126-6
  2. 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
  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.
  4. T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, 1991.
  5. W. K. Nicholson, Introduction to Abstract Algebra, PWS, Boston, 1998.

피인용 문헌

  1. UNIT-DUO RINGS AND RELATED GRAPHS OF ZERO DIVISORS vol.53, pp.6, 2016, https://doi.org/10.4134/BKMS.b150684
  2. Structure of Abelian rings vol.12, pp.1, 2017, https://doi.org/10.1007/s11464-016-0586-z