REGULAR ACTION IN Zn
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초록
Let n be any positive integer and Zn = {0, 1, . . . , n−1} be the ring of integers modulo n. Let Xn be the set of all nonzero, nonunits of Zn, and Gn be the group of all units of Zn. In this paper, by investigating the regular action on Xn by Gn, the following are proved : (1) The number of orbits under the regular action (resp. the number of annihilators in Xn) is equal to the number of all divisors (̸= 1,n) of n; (2) For any positive integer n, g∈Gn g ≡ 0 (mod n); (3) For any orbit o(x) (x ∈ Xn) with |o(x)| ≥ 2, y∈o(x) y ≡ 0 (mod n).
Abstract
Let n be any positive integer and Zn = {0, 1, . . . , n−1} be the ring of integers modulo n. Let Xn be the set of all nonzero, nonunits of Zn, and Gn be the group of all units of Zn. In this paper, by investigating the regular action on Xn by Gn, the following are proved : (1) The number of orbits under the regular action (resp. the number of annihilators in Xn) is equal to the number of all divisors (̸= 1,n) of n; (2) For any positive integer n, g∈Gn g ≡ 0 (mod n); (3) For any orbit o(x) (x ∈ Xn) with |o(x)| ≥ 2, y∈o(x) y ≡ 0 (mod n).
참고문헌 (3)
- J. A. Cohen, "Half-transitive group actions in a compact ring", J. Pure Appl. Algebra, 60, pp.139~153 (1989)
- J. Han, "Regular action in a ring with a finite number of orbits", Comm. Algebra, 25(7), pp.2227~2236 (1997)
- J. Han, "Group actions in a unit-regular ring", Comm. Algebra, 27(7), pp.3353~3361 (1999)