半单算子与半单类

THE SEMISIMPLE OPERATOR AND SEMISIMPLE CLASSES

1981年,W.G.Leavitt 深入地讨论了上根算子与根环类之间的关系(文〔1〕).但对于半单算子尚未见文献论述。本文考察了任一环类在半单算子作用下的状况。文中引进的(?)-可半单类在某种意义下可认为是〔1〕中 r-类的对偶;给出了(?)可半单类的刻划并利用上正则类考察了下根 L(M)的半单类。文中还给出半单算子作用下的环类 (?)M 成为遗传根和超幂零根的半单类的充要条件;得到了 U(?)M 是超幂零根的一个充分条件。

The upper radical operator and the semisimple operator are respectvelyfundamental tools in constructing radical classes and semisimple Classes.W.G.Leavitt and other authors have made a deep study of the upper radicaloperator.In this paper we make an investigation on the semisimple operator.In §1 the (?)-semisimple class is defined and characterized which is dual tothe r-class in [1] in some sense.In §2,the sufficient and necessary condi-tions are given for (?)M being the semisimple class of an hereditary radicaland a supernilpotent radical in radical theory.