其上任一模都是循环模直和的环

Rings on which any Module Is a Direct Sum of Cyclic Modules

本文在双环的前提下,用任一模都是循环模直和这一模特征,对某类环进行了完全刻划.得到了主要定理:设R是有1的双环.那么下列等价:(α) R上任一左模都是循环模直和;(b) R是左Artin主理想环;(c) R是左Noether环,并且对R的任一理想I,R/I是(左) 自内射环.并且还进一步得到,一个环如果是局部环直和,那么上述(C)成立蕴含着这个环一定是双环.

In this paper we have studied the rings on which any module is a direct sum of cyclic modules. The main conclusion is Theorem: Let R be a double ring. Then the following conditions are equivalent: ( a) any R-module is a direct sum of cyclic modules, (b) R is a Artinian principal ideal ring. (c) R is Noetherian and, for any ideal Iof the ring R/ I is self-injective.