On sum of a nilpotent and an ideally finite algebras

We study associative algebras $R$ over arbitrary fields which can be decomposed into a sum $R=A+B$ of their subalgebras $A$ and $B$ such that $A^{2}=0$ and $B$ is ideally finite (is a sum of its finite dimensional ideals). We prove that $R$ has a locally nilpotent ideal $I$ such that $R/I$ is an extension of ideally finite algebra by a nilpotent algebra. Some properties of ideally finite algebras are also established.