σ—根与σ—半单类

σ-RADICALS AND σ-SEMISIMPLE CLASSES

定义σ-根和σ-半单类并给出它们的特征,得到根与半单类的新刻划。

A relation σ of associative rings is called an H-Relation if σ satisfies the proper- ties:(1)IσR implies I is a subring of R.(2)if IσR and (?) is a homomorphism of R,then and J is an ideal of R, then . (4)IσR whenever I is an ideal of R. A radical class r is defined a σ-radical if for any ring R and IσR and Ier, the ideal I-R of r generated by I is in r. In this paper the following results are obtained: Theorem 1 Let σ is a H-relation, a rings class r is a σ-radical if and only if r satisfies: (i)r is hoinomorphically closed. (ii) if every onzero homomorphic image of a ring R contain a nonzero σ-subring in r, the itself is in r. As a straight Consequence, theorem 1 positively answer the open problem 1 raised in F.A.Szasz's'radicals of rings' in another mathod, thus we obtain a new characterization of radical class. Theorem 2 A rings class S is in a σ-Semisimple class for an σ-radical if and only if s satisfies: (i) everynonzero σ-subring of a.ring R in s can be homomrphically mapped on to a nonzero ring from s. (i) if every nonzero σ-subring of a ring R can be homomorphically mapped on to a nonzero ring from s, then R belong to the class s. Theorem 3 A rings class s is a semisimple class if and only if s satisfies; (s1) if R∈s, then every nonzero accessible subring of R has a nonzero homomorphic image in s. (s2) if every nonzero accessible subring of R has a nonzero homomorphic image in s, then R∈s.