关于环上自由模与矩阵环的讨论

Discussion of Free Running Model on the Ring and Matrix Ring

环上的自由模是域上线性空间的一种推广,因而线性空间的许多性质可以自然地推广到环上的自由模.文[1]指出,交换环上自由模的基所含元素的个数是自由模的一个不变量,即基元个数不变性.这里对任意环上自由模的基及相关矩阵进行了讨论,给出了任意环上两个自由模R(m)与R(n)同构的充要条件,R(m),R(n)分别是秩为m,n的自由R模,并且Hom(R(m),R(n))是秩为mn的自由R-模,同时做出了使R(m)≌R(n)、但m=n不成立的反例.

Free running model on the ring is an expansion of linear vector space in fields. Therefore, many features in linear vector space can be naturally used free running model on the ring. As [ 1], it points out that the basic element number in radicals of free running model is an invariant in commutative ring, that is to say, the invariance of basic element number. Radicals on the arbitrary rings of the free running models are discussed. The sufficient conditions of isomorphism of the two running models R and R on the arbitrary rings are given; R and R ranks are m, n free R-model respectively. Besides, ranks of Hom （R, R） are rn n free running models, meanwhile, example of R R is given, but the formula of mn doesn＇t hold water.