遗传上根和超幂零根与其补根成偶的条件

设M是一个环类,令∪M={A|(?)O≠A/I,A/I(?)M}.H∪M={A|(?)O≠I△△A,(?)O≠A/J,I/J(?)M}本文证明了当M是r一环类时,H∪M是含在∪M中的最大遗传根,我们称之为由M决定的遗传上根。利用遗传上根,我们给出了超幂零根S与补根S′形成根偶的几个充要条件。

The Hereditary Upper Radicals and The Conditions Which the Supernillpotent Radical and Its Supplementary Radical Combine Dual of Radicals

Let M be a class of rings ∪M= {A|for (?)A/I≠O, A/I(?) M} ,SM = {A|(?)O≠I△ A,I(?)M} ,H∪M={A|(?)O≠I△△A,(?)I/J≠O,I/J(?)M}.Then we obtained some results;Theorem 1. Let M bear-Class (i.e. for ecery o≠A/I∈S∪M), Then. H∪M is the Hereditary upper Radical that is determined by the class M.(i.e.the maximal hereditary radical in ∪M).Theorem 2. Let S be Supernilpetent. S' , Supplementarg radicall of S, Then S and S' combine dual pair of radicals iff S = H∪K1, iff S = H∪K2; where K1={A|A is S-semisimple, and for every non-zero homorphism image A/I of A,A/I contains a non-zero strongy S-semisimple ideal} K2={A|for erery non-zero accessible subring I of A , I contains a non-zero strongly S-semisimple idral} .