与线性变换的完全环同构的环理论(Ⅱ)

<正> 为了进一步对本原环结构的研究,本文引进规范环的概念,我们说环R是规范的,若R是一个线性变换完全环并且及的基座对于任一对应基{E_i}皆有=∑RE_i=∑E_iR.容易知道,满足单侧理想极小条件的单纯环必是规范的.

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

A ring Ω is called a complete ring of linear transformations if there exists a vector space A over a division ring such that Ω is the ring of all linear transformations of A Let be the socle of Ω. Then a subset {E_i} r of is called a correlative basis of if and only if either where, E_i~2=E_i,E_iE_j = 0, i≠j, i, j ∈F,Definition 1. A ring Ω is called a normal ring if and only if Ω is the complete ring of all linear transformations and for every correlative basis {E_i} r of.Definition 2.A primitive ring R is called normalizable if and only if there exists a normal ring Ω such that R Ω and the socle of Ω contains that of R.Definition 3. A chain of rings Ω_o Ω_1 …Ω_a …is called a standard ascending chain if it satisfies the following conditions:(i) for every ordinal number a there corresponds a component Ω_a of this chain.(ii) every component Ω_a of this chain is the complete ring of all linear transformations such that Ω_a is dense in Ω_(a+1).(iii) let be the socle of Ω_a, then …(iv) if a<β then Ω_βL_α is a minimal left ideal of Ω_β for every minimal left ideal L_a of Ω_a.Definition 4.A standard ascending chain of rings Ω_o Ω_1 …Ω_a… is called a non-set chain if Ω_a is a non-set, where ∪ is the set sum.Definition 5.A primitive ring R is called a ring having a non-set standard chain of rings Ω_o Ω_1 …Ω_a …if this chain is a non-set standard chain of rings and R is dense in Ω_o.Theorem Ⅰ. Any primitive ring is either a normalizable ring or a ring having a non-set standard chain of rings.Definition 6. A matrix ring M over a division ring K is called a matrix ring having a finite number of columns with non-zero eatries if and only if for every element r of M there corresponds n(r) number of cohunns of r such that all entries outside these n(r) columns are zeros, where n(r) is a positive integer.Theorem Ⅱ; Let R be a normalizable primitive ring, and I be any subset of elements of the soele of R with cardinal number Then the subring generated by I is isomorphic to a subring of matrix ring having a finite number of columns with non-zero entries over a division ring.