Integrality of subrings of matrix rings

Lance W. Small and Adrian R. Wadsworth

Vol. 116 (1985), No. 1, 195–200

DOI: 10.2140/pjm.1985.116.195

Abstract

Let A ⊆ B be commutative rings, and Γ a multiplicative monoid which generates the matrix ring M_n(B) as a B-module. Suppose that for each γ ∈ 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 A, the integral closure of A∕P is completely integrally closed, then the algebra A(Γ) 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(Γ) is a finitely generated A-module.

Mathematical Subject Classification

Primary: 16A38, 16A38

Milestones

Received: 2 June 1983

Published: 1 January 1985