根论中的S-单环

本文引入S-单环的概念,给出S-单环类的若干性质,把S-单环用于特殊根格、超幂零根格的结构的讨论,考虑了原子问题,给出1]中问题10的部分回答。

S-simple Rings in the Radical Theory

Let r denote a radical class, S a semisimple class determied by r. We defined that the ring R is S-simple ring if R∈S and R/I∈S for every non-zero proper ideal I of R. Let Sr denote the class of all the S-simple rings, A the class of all the associative rings, β the Baer lower nil radical. We say that the supernilpotent radical r_1 covers r_2 if r_1 r_2 and there does not exsit a supernilpotent radical r such that r_1 r r_2. Let M A, We denote IM={R∈A| EA∈M, EI A, ∈R I} IM={R∈A | AR/I ≠0, ED/I (≠0) ∈M_ D/I R/I} LA=L(r∪I(A)) Theorem14. If r is a supernilpotent radical, A(≠0)∈Sr, then L_A covers r in the lattice of all the supernilpoteht radicaIs. Corollary16. If A(≠0) ∈S_3, then L_A is an atom in the supernilpotent radical lattice. Thus we answered the problem 10 of 1] in part.