On algebras that are sums of two subalgebras

Abstract

We study an associative algebra \(A\) over an arbitrary field \(K\) that is a sum of two subalgebras \(B\) and \(C\) (i.e. \(A=B+C)\). Let \(\mathcal{M}\) be the class of algebras such that \(B, C\in \mathcal{M}\) implies \(A\in \mathcal{M}\). We prove, under some natural additional assumptions on \(\mathcal{M}\), that if \(B\) and \(C\) have ideals of finite codimension from \(\mathcal{M}\), then \(A\) has an ideal of finite codimension from \(\mathcal{M}\), too. In particular we show that if \(B\) and \(C\) have left T-nilpotent ideals (or nil \(PI\) ideals) of finite codimension, then \(A\) has a left T-nilpotent ideal (or nil \(PI\) ideal) of finite codimension.