关于U－循环环和双循环环

设Ｒ是有单位元的环．我们称Ｒ为循环环，如果加群（Ｒ，＋）是循环群；称Ｒ为Ｕ－循环群，如果Ｒ的全体单位作成的乘群Ｕ（Ｒ）是循环群；称Ｒ为双循环环，如果（Ｒ，＋）和Ｕ（Ｒ）都是循环群．本文利用（Ｒ，＋）与Ｕ（Ｒ）的一些性质讨论环Ｒ的性质和结构，所得主要结果如下：（１）若Ｒ是Ａｒｔｉｎ半单环，则Ｕ（Ｒ）是有限的当且仅当Ｒ是有限的．（２）域Ｆ是Ｕ－循环环当且仅当Ｆ是有限的．（３）若Ｒ是域Ｆ上所有ｎ阶上三角形矩阵作成的环，则Ｒ是Ｕ－循环环当且仅当ｎ＝２和Ｆ≌Ｚ２．（４）若Ｒ是无限环，则Ｒ是双循环环当且仅当Ｒ≌Ｚ．（５）设Ｒ是有限环且｜Ｒ｜＝ｎ＞１，则Ｒ是双循环环当且仅当Ｒ≌Ｚｎ，ｎ为２，４，ｐｋ，２ｐｋ，其中ｐ为任意奇素数，ｋ为任意正整数．

On the U-Cyclic Rings and Double Cyclic Rings

Suppose that R is a ring with unit element. We denote the additive group of R by (R , + ) ,and the unit group of R by U(R). We call R is a cyclic ring if (R, + ) is a cyclic group, R a Ucyclic ring if R is a cyclic group, and R a double cyclic ring if (R, + ) and U(R) both are cyclic groups. In this paper, we discuss the property and the structure of R by use of (R , + ) and U (R). The main results are as follows.1. Let R be an Artin Semi-simple ring. Then U(R) is finite if and only if R is finite.2. The field F is a U-Cyclic ring if and only if F is finite.3. Let R be the ring of n ×n(n>1) upper-triangular matrices with entries from the field F.Then R is a U-cyclic ring if and only if n=2 and F≌Z2.4. Let R be an infinite ring. Then R is double Cyclic ring if and if only if R≌Z.5. Let R be a finite ring and |R|=n>1. Then R is a double cyclic ring if and only if R≌Zn, where n are the integers 2, 4, pk, 2pk (p is any odd prime number, and k any positive integer).