IMCS/Publications/BASM/Issues/BASM n.2 (30), 1999/

On the intersection of radical classes. (English)

Authors: Sands A. D.


If the intersection of radical classes is taken then a radical class is obtained. However it is not necessarily the case for a given ring $R$ that the radical of $R$ obtained in this way is the intersection of the radicals of $R$ corresponding to the original radical classes. In this paper we consider this problem for the intersection of two radical classes. We obtain chains of ideals of $R$ associated with the two radicals. By means of examples we show that the radical corresponding to the intersection can occur at any point in each such chain. One problem where there is a difference between considering the intersection of the radical classes and the intersection of the ideals is that of pseudo-complements. For radicals $ \alpha , \gamma , \theta $ R.L.Snider uses the relationship $ \alpha (R) \cap \gamma (R) \subseteq \theta (R) $ for each ring $R$. He obtains positive results whenever $ \alpha , \theta $ are hereditary radicals. Here we consider instead the weaker relationship $ \alpha \cap \gamma \subseteq \theta $. Positive results are obtained whenever $ \alpha $ satisfies a condition somewhat weaker than hereditary while no restriction need be placed on $ \theta $. Examples are given to illustrate the difference even in the hereditary case.

Departament of Mathematics
Dundee University, Dundee DD1 4HN, Scotland