Representability of algebras finite over their centers

Any (associative) left Noetherian algebra over a field, which is finitely generated as an algebra over a central subring, is representable. As a special case, we give a short proof of the well-known theorem that any algebra over a field, which also is finite (as a module) over a central affine algebra, is representable. We give a counterexample to some natural generalizations, as well as a conjectured generic counterexample, together with specific positive results for irreducible algebras and finitely presented algebras. Also, based on joint work of the second author with Amitsur (posthumous), we show that any semiprimary PI-algebra with radical squared 0 is weakly representable.