Extended Centres of Finitely Generated Prime Algebras

Abstract

Let K be a field and let A be a finitely generated prime K-algebra. We generalize a result of Smith and Zhang, showing that if A is not PI and does not have a locally nilpotent ideal, then the extended centre of A has transcendence degree at most GKdim(A) −2 over K. As a consequence, we are able to show that if A is a prime K-algebra of quadratic growth, then either the extended centre is algebraic over K or A is PI. Finally, we give an example of a finitely generated non-PI prime K-algebra of GK dimension 2 with a locally nilpotent ideal such that the extended centre has infinite transcendence degree over K.