On rings which are sums of subrings

Abstract There are presented some generalizations and extensions of results for rings which are sums of two or tree subrings. It is provided a new proof of the well-known Kegel’s result stating that a ring being a sum of two nilpotent subrings is itself nilpotent. Moreover, it is proved that if R is a ring of the form $R=A+B$ , where A is a subgroup of the additive group of R satisfying $A^d\subseteq B$ for some positive integer d and B is a subring of R such that $B\in S$ , where S is N -radical contained in the class of all locally nilpotent rings, then $R\in S$ .