Value Functions and Dubrovin Valuation Rings on Simple Algebras

TL;DR

This paper explores the relationship between Dubrovin valuation rings and gauges in simple algebras, establishing conditions under which certain subrings are minimal gauges and connecting valuation theory with noncommutative algebra structures.

Contribution

It demonstrates the equivalence between minimal v-gauges and intersections of Dubrovin valuation rings, extending valuation theory to noncommutative central simple algebras.

Findings

Characterization of minimal gauges in simple algebras.

Existence results for minimal gauges.

Construction of gauges from valuations on central simple algebras.

Abstract

In this paper we prove relationships between two generalizations of commutative valuation theory for noncommutative central simple algebras: (1) Dubrovin valuation rings; and (2) the value functions called gauges introduced by Tignol and Wadsworth in [TW1] and [TW2]. We show that if v is a valuation on a field F with associated valuation ring V and v is defectless in a central simple F-algebra A, and C is a subring of A, then the following are equivalent: (a) C is the gauge ring of some minimal v-gauge on A, i.e., a gauge with the minimal number of simple components of C/J(C); (b) C is integral over V with C = B_1 \cap ... \cap B_xi$ where each B_i is a Dubrovin valuation ring of A with center V, and the B_i satisfy Graeter's Intersection Property. Along the way we prove the existence of minimal gauges whenever possible and we show how gauges on simple algebras are built from gauges on central simple algebras.