满足主理想极小条件的环

RINGS SATIFYING THE MINIMUM CONDITION ON PRINCIPAL IDEALS

本文证明了满足主理想极小条件的环同时满足有限生成理想极小条件;证明了有单位元素的环R满足主理想极小条件的充要条件是矩阵环R_n满足主理想极小条件;证明了这类环的反单根、Levitzki根以及Bear下根相重,顺便对Szasz问题100做出肯定回答。

In this paper, we prove that every ring satifying the minimum condition on principal idcals satifies the minimum condition on finitely generated idcals, and that every ring A with a unity element satifies the minimum condition on principal idcals if and only if the matrix ring A_n satifies the minimum condition on principal ideals. We prove yet that the antisimple radical, the Levitzki radical and the Bear lower radical of this type rings are coincident. In addition, we give a positive answer for Szasz's question 100.