On rings which are sums of two subrings
A result proved by \textit{O. H. Kegel} [Math. Ann. 149, 258-260 (1963; Zbl 0106.254)] states that a ring \(R\) is nilpotent if it is a sum of two nilpotent subrings \(R_ 1\) and \(R_ 2\). In this paper we study the case when one of the subrings is nilpotent (\(T\)-nilpotent) and the other is S- radical, for several radicals \(S\), including the most popular radicals. We get, for example, that if \(R_ 1\) is nilpotent and \(R_ 2\) is S- radical, where \(S\) is an \(N\)-radical, then \(R\) is \(S\)-radical. Also if \(R_ 1\) is right \(T\)-nilpotent and \(R_ 2\) is Jacobson radical (locally nilpotent), then \(R\) is Jacobson radical (locally nilpotent). The analogous results for the nil radical are equivalent to the Koethe conjecture. Several other results of this type and examples are included in the paper. Finally, it is an open question whether \(R\) is prime radical if \(R_ 1\) is \(T\)-nilpotent and \(R_ 2\) is prime radical.
scientific article; zbMATH DE number 4051819