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REGULAR ACTION IN ℤn

  • 투고 : 2016.12.12
  • 심사 : 2017.02.13
  • 발행 : 2017.05.31

초록

Let n be any positive integer and ${\mathbb{Z}}_n=\{0,1,{\cdots},n-1\}$ be the ring of integers modulo n. Let $X_n$ be the set of all nonzero, nonunits of ${\mathbb{Z}}_n$, and $G_n$ be the group of all units of ${\mathbb{Z}}_n$. In this paper, by investigating the regular action on $X_n$ by $G_n$, the following are proved : (1) The number of orbits under the regular action (resp. the number of annihilators in $X_n$) is equal to the number of all divisors (${\neq}1$, n) of n; (2) For any positive integer n, ${\sum}_{g{\in}G_n}\;g{\equiv}0$ (mod n); (3) For any orbit o(x) ($x{\in}X_n$) with ${\mid}o(x){\mid}{\geq}2$, ${\sum}_{y{\in}o(x)}\;y{\equiv}0$ (mod n).

키워드

참고문헌

  1. J. A. Cohen and K. Koh, Half-transitive group actions in a compact ring, J. Pure and Appl. Algebra 60 (1989), 139-153. https://doi.org/10.1016/0022-4049(89)90126-6
  2. J. Han, Regular action in a ring with a nite number of orbits, Comm. Algebra 25 (1997), no.7, 2227-2236. https://doi.org/10.1080/00927879708825984
  3. J. Han, Group actions in a unit-regular ring, Comm. Algebra 27 (1999), no.7, 3353-3361. https://doi.org/10.1080/00927879908826632