  This note concerns the still open question of representability of Noetherian
PI-algebras. Extending a result of Rowen and Small (with an observation of
Bergman) that every finitely generated module over a commutative Noetherian
ring containing a field is representable, we provide a representability
machinery for a Noetherian PI-algebra $R$ containing a field, which includes
the case that $R$ is finite (as a module) over a commutative subalgebra
isomorphic to $R/N$. We construct a family of non-representable PI-algebras
demonstrating the sharpness of these results, as well as of some well known
previous representability results.
