Integrality of subrings of matrix rings

Abstract

Let A c B be commutative rings, and Γ a multiplicative monoid which generates the matrix ring Mn{B) as a ^-module. Suppose that for each γ e Γ its trace tr(γ) is integral over A, We will show that if A is an algebra over the rational numbers or if for every prime ideal P of A9 the integral closure of A/P is completely integrally closed, then the algebra A(T) generated by Γ over A is integral over A. This generalizes a theorem of Bass which says that if A is Noetherian (and the trace condition holds), then A(T) is a finitely generated A -module. Our generalizations of the theorem of Bass [B, Th. 3.3] yield a simplified proof of that theorem. Bass's proof used techniques of Procesi in [P, Ch. VI] and involved completion and faithfully flat descent. The arguments given here are based on elementary properties of integral closure and complete integral closure. They serve also to illuminate a couple of theorems of A. Braun concerning prime p.i. rings integral over the center. One might expect that integrality of tr(γ) for γeΓ would be sufficient to assure that ^4(Γ) is integral over A. But this is not so, as we will show with a counterexample . As it frequently happens with traces, complications arise in prime characteristic.