global dimension of a ri ng

In this paper,the author discusses the order of an element of periodic groups and free groups,studies the order of an element of the additive group of a ring and establishes the concept of the characteristic number of a ring,and gives two popularizations of the order of an element of groups:the order of an element to a subgroup and the order of an element of semigroups.

　讨论了周期群与自由群的元素的阶,研究环的加群的元素的阶并建立环的特征数的概念。