<正> 熟知地,满足极小条件的单纯环只与一个有限维向量空间的线性变换的完全环同构.并且此向量空间如取为左向量空间的话,那末R的任一极小右理想均可取为此左向量空间.在没有有限条件情况下,Jacobsoo用本原环来取代这种单纯环.接着Wolfson研

An abstract ring which is isomorphic to the ring of all linear transformations of a vector space has been studied by wolfson. In this paper we follow his work and obtain a main theorem. Before formulating we first introduce the following.Definition: Let M be a (F, R)-module, i.e. M is a left F, right R-module, and let M' be a (K, R)-module, where F, K, R are rings. We say that M and M' are (ψ, I)isomorphic if there exists a mapping S satisfying the following conditions:(i) S is an isomorphism of (M, +) onto (M', +).(ii) There exists an isomorphism ψ of F onto K such that for all x ∈M, r ∈R we hav.e (fx)S = f~ψ(xS), (fxr)S = f~ψ(xS)r.Theorem: Let R be an abstract ring. Then R is isomorphic to the ring of linear transformations of a left vector space A over a division ring F if and only if the following conditions are satisfied:(i) R has a socle with and every non-zero ideal of R contains.(ii) Suppose that S_1⊕S_2,S_1 are minimal left ideals Of R, then(iii) R has an identity.Moreover, if R satisfies the above three conditions then every minimal right ideal A'= eR of R can be expressed as a left vector space over the division ring K and R is also the complete ring of K-linear transformations of A'. Furthermore there exists a (ψ, I)-isomorphism of A as (F, R)-module onto A' as (K, R)-module, and the rank of A is equal to that of Thus the left vector space of R except semi-linear transformation is uniquely determined by the soele.