与线性变换的完全环同构的环理论(Ⅳ)

<正> 基座概念对本原环的结构研究起着十分重要作用.为了对本原环的结构作进一步研究,我们引进俨基座概念.通常基座概念就是我们特殊情形的o-基座概念.利用ν-基座概念,我们建立了ν-结构定理。通常本原环结构定理(见2]p.75)是ν-结构定理的一种特殊情况. 为了引进ν-基座,我们改变一下本原环的基座定义,使它具有能表达ν-基座的一般形式的特点且能建立所要求的ν-结构定理.为此我们来提一下§2中所获得的结果:

A THEORY OF RINGS THAT ARE ISOMORPHIC TO THE COMPLETE RINGS OF LINEAR TRANSFORMATIONS(IV)

Let R be a primitive ring, A be a faithful irreducible R-module. Suppose that F be the centralizer of R-module A, then it is clear that A Fu_i is a vecter space over F. Let Ω be the complete ring of linear transformations of A and let T_ν be the set of all linear transformations of A such that their ranks are all<.Then we can formulate the following definition.Definition: An ideal of R is said to be a v-socle if and only if satisfies the following conditions:(iv)let{u_i}r be a basis,{E_i}r be the set of Ω satisfying the conditions u_iE_j=δ_(ij)u_i,i,j∈Γand let l∈T_v,In this paper we have proved that the o-socle of R defined above is equivalent to the socle of R in the usual meaning.Furthermore, we introduce in this paper the concepts of v-topology,v-A'-summable dual vector space (A, A') and v-total A' too. Then we can obtain the following theorems:Theorem Ⅰ: Let (A,A') be a v-A'-summable dual vector space over F, A' be v-total and let (A) be the set of all continuous linear transformations (by the v-A'-topology), denote (A)=T_v(A). Suppose that R is isomorphic to a subring of (A), which includes (A), then R includes the v-socle.Theorem Ⅱ: Let R be a primitive ring with v-socle. Then there exist a v-A'-summable dual vector space (A,A') over F, A' is v-total and R is isomorphic to a subring of L(A), which includes T(A), where L(A) denotes the ring of all strong continuous linear transformations(by the v-A'-topology), T(A)=T_vL(A).If we set v=0, then it is clear that the o-topology introduced above is namely the usual finite topology and the theorems Ⅰ and Ⅱ are obviously consistent with the usual socalled "Structure Theorem".