Integral extensions of noncommutative rings

In this paper we study integral extensions of noncommutative rings. To begin, we prove that finite subnormalizing extensions are integral. This is done by proving a generalization of the Paré-Schelter result that a matrix ring is integral over the coefficient ring. Our methods are similar to those of Lorenz and Passman, who showed that finite normalizing extensions are integral. As corollaries we note that the (twisted) smash product over the restricted enveloping algebra of a finite dimensional restricted Lie algebra is integral over the coefficient ring and then prove a Going Up theorem for prime ideals in these ring extensions. Next we study automorphisms of rings. In particular, we prove an integrality theorem for algebraic automorphisms. Combining group gradings and actions, we show that if a ring R is graded by a finite group G, and H is a finite group of automorphisms of R that permute the homogeneous components, with the order of H invertible in R, then R is integral over R ₁ ^H, the fixed ring of the identity component. This, in turn, is used to prove our final result: Suppose that if H is a finite dimensional semisimple cocommutative Hopf algebra over an algebraically closed field of positive characteristic. If R is an H-module algebra, then R is integral over R ^H, its subring of invariants.