Note on Algebras that are Sums of Two Subalgebras Satisfying Polynomial Identities

We study an associative algebra A over an arbitrary field, that is a sum of two subalgebras B and C (i.e. A = B+C). We prove that if B has a nil ideal of bounded index, and that C has a commutative ideal, both of finite codimension in B and C, respectively, then for some nil PI ideal I of A the ring A/I has a commutative ideal of finite codimension.