Abstract:

In this paper, we study (Schelter) integral extensions of PI-rings.  We prove in particular lying over, going up and incomparability for prime ideals.  A major result is transitivity of integrality: If R \subseteq S \subseteq B are PI-rings such that B is integral over S and S is integral over R, then B is integral over R.  Next, we obtain a powerful criterion for integrality: If S is a prime PI-ring such that its center is integral over a Noetherian subring R of S, then S is integral over R.  This allows interesting applications to the theory of finite group actions.  Further topics concern Eakin-Nagata type results and embeddings of quotient rings for integral extensions. Finally, we analyze the relationship between module-finite extensions and finitely generated integral extensions, obtaining positive results for affine Noetherian PI-algebras and algebras satisfying certain restrictions on PI-degrees (e.g., algebras of low PI-degree).