Optimal representability results of PI-algebras: nilpotency, growth and chain conditions (Q6551457)

An affine (that is finitely generated) algebra \(A\) over a field \(F\) is representable if it can be embedded into a matrix algebra over a field, or more generally, over a commutative ring. (The authors adopt the term \textit{weakly representable} for the latter.) It is well known that if \(A\) is affine the two notions of representability coincide. By the well known Amitsur and Levitzki theorem, every such algebra is PI. The authors study how conditions imposed on ideals of \(A\) affect the representability of \(A\). First they consider \(A\) Noetherian with nilradical \(N\), and containing a commutative subalgebra \(C\) such that \(R/N\) is a finite module over \(C+N/N\). They prove that in this situation \(A\) is representable (theorem 1.1).\N\NAn algebra (or a ring) \(A\) is semiprimary whenever \(A/N\) is semisimple Artinian. There is no general result concerning the representability of Artinian PI algebras. Here the authors prove that for any field there exists a semiprimary PI algebra that is not representable. In the example they construct it holds that \(\dim N^2 = 1\) and moreover \(N^3=0\).\N\NFor the entire collection see [Zbl 1539.11006].