Optimal representability results of PI-algebras: nilpotency, growth and chain conditions

We investigate representability results for PI-algebras satisfying various chain conditions and examine their optimality. We show that a Noetherian PI-algebra, whose quotient by the radical is finite over its center, is representable. We derive several applications and give a counterexample to this representability result when Noetherianity is relaxed to ACC on ideals, even within the class of affine algebras of linear growth. This example has radical cubed zero, and we show that if the radical squared is zero then such a counterexample cannot exist. We then modify our construction to yield a non-representable semiprimary PI-algebra with radical cubed zero, proving the tightness of a result of Amitsur, Rowen and Small, which was left open by Rowen and Small. We conclude with a discussion of the remaining open problems.