On radicals of rings which are sums of two subrings

摘要

Rings which are sums of two subrings were studied in many papers (cf. [2, 4, 6, 7, 8]). There are many open questions in the area among them the famous Koethe problem (in [4] it was proved that the Koethe problem is equivalent to the problem of whether a ring which is the sum of a nil subring and a nilpotent subring must be nil). In this paper we continue the study of radicals of such rings. In particular we extend several results of [4] and [6] and obtain some new results concerning the open question of whether a ring which is a sum of two prime radical subrings must be prime radical.