Radicals Coinciding with the Jacobson Radical on Linearly Compact Rings

Abstract

It is the purpose of this paper to approximate or to localize the Jacobson radical class by upper and lower radical classes in the following sense: we shall construct two radical properties which will coincide with the Jacobson radical on the class of linearly compact rings such that if a radical coincides with the Jacobson radical on the linearly compact rings, then this radical class lies between the given radical classes. In [2] Divinsky has given boundaries for radicals coinciding with the Jacobson radical on the class of artinian rings. Similar investigations were done by F. Sziasz [9] on the class of the so called MHR-rings. Since the class of MHR-rings is a biger one than that of the artinian rings, so the boundaries are closer. Here we present similar investigations according to the class of linearly compact rings, and it will turn out that our approximation is better than those made earlier, it will be proved that neither the Brown-McCoy radical, nor Koethe’s nil-radical (and so also Baer’s lower radical) do not coincide with the Jacobson radical on the class of linearly compact rings. At the end of the paper we give some suggestions for further researches in these topics.