The hereditariness of the upper radical

Summary Starting from a regular classM, one can construct the upper radicalU M of the classM in a category which is like that of associative, alternative or not necessarily associative rings, or that of Lie rings. It turns out that in quite a few cases the upper radical is hereditary. (cf.Suliski 7], Rjabuhin 6], Armendariz 2], Szász—Wiegandt 8]).W. G. Leavitt has suggested the problem: Give a necessary and sufficient condition to be satisfied by the regular classM so that the upper radical classU M ofM is hereditary. In the present paper we shall give such a necessary and sufficient condition. If the classM satisfies an even stronger condition, then theU M-semisimple objects are subdirectly embeddable in a (direct) product ofM-objects. Also a necessary and sufficient condition is given which assures that eachU M-semisimple object can be subdirectly embedded in a (direct) product ofM-objects.This work was done when the second named author was in the University of Islamabad under UNESCO-UNDP Special Fund Pak. 47.