环的广直和

THE HYPERDIRECT SUMS OF RINGS

直和是研究代数系统结构的重要工具。利用离散直和可以刻划某些具有一定条件的环类,利用亚直和可以刻划一般的半单环类。本文引进一种介于离散直和与亚直和之间的直和概念,利用它给出了弱双正则环的一个结构定理,描述了双正则根环类、么单根环类的结构,并对一般环得到了它们关于这两个根的一种亚直和分解。

Let A_α,α∈W,be a family of rings with the nonempty index set W.A subringA of the complete direct sum ΠA_α is called a hyperdirect sum of rings A_α if itcontains the discrete direct sum ΣA_α.We have proved the following theorems:A nonzero weakly biregular ring can be expressed as a hyperdirect sum ofrings with the unit.A ring A is a prebiregular ring if and only if it is a hyperdirect sum of thesimple rings with the unit.In the last,two theorems on the decomposition of a ring A with respect toits biregular radical and simply unital radical are proved.