关于Olson和Jenkins定义的根类

On the Radical Classes Defined by Olson and Jenkins

ＤｗｉｇｈｔＭ．Ｏｌｓｏｎ和ＴｅｒｒｙＬ．Ｊｅｎｋｉｎｓ［４］定义了由任意环类确定的一种根类（Ｍ）．一个环Ｒ∈（Ｍ）当且仅当Ｒ的每一个非零同态象或者包含一个非零Ｍ理想或者有本质理想．他们提出两个需要进一步讨论的问题．对于一个同态闭环类Ｍ，（Ｍ）和由Ｍ生成的低根类（Ｍ）有什么关系？对于一个正则环类Ｍ，是否有环类Ｎ使得上根（Ｍ）＝（Ｎ）？本文将对这两个问题给出回答并讨论这种根类簇的性质．

Dwight M. Olson and Terry L. Jenkins[4] defined one kind of radical classes(M) by an arbitrary class M of rings. A ring Re,(M) iff each nonzero homomorphic image of R either contains a nonzero M-ideal or has an essential ideal. They suggested two problems for further investigation:For a homomorphicaly closed class M of rings,what is the relationship between(M) and the lower radical class (M) generated by M? For a regular class M of rings,is there any way to realize ac(M) as(M) for some class N of rings related to M? The two problems are given solutions and discussed the properties of the collection of all such radical classes.