On sum of a nilpotent and an ideally finite algebras

Abstract

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.