Some remarks on growth of algebras over commutative noetherian rings

摘要

Using a growth function,GKdefined for algebras over integral domains, we construct a generalization of Gelfand Kirillov dimensionGGK.GGKcoincides with the classical no-tion ofGKfor algebras over a field, but is defined for algebras over arbitrary commutative rings. It is proved thatGGKexceeds the Krull dimension for affine Noetherian PI algebras. The main result is that algebras ofGGKat most one are PI for a large class of commutative Noetherian base rings including the ring of integers, Z. This extends the well-known result of Small, Stafford, and Warfield found in [11].