Monomial, Gorenstein and Bass Orders

TL;DR

This paper investigates monomial orders in central simple algebras over local fields, providing criteria for when they are Gorenstein or Bass, and characterizing Bass monomial orders as hereditary or Eichler orders of period two.

Contribution

It introduces a criterion for monomial orders to be Gorenstein or Bass and characterizes Bass monomial orders explicitly.

Findings

A criterion for monomial orders to be Gorenstein.

A criterion for monomial orders to be Bass.

Bass monomial orders are exactly hereditary or Eichler orders of period two.

Abstract

In this article we study a class of orders called {\it monomial orders} in a central simple algebra over a non-Archimedean local field. Monomial orders are easily represented and they may be also viewed as a direct generalization of Eichler orders in quaternion algebras. A criterion for monomial orders to be Gorenstein or to be Bass is given. It is shown that a monomial order is Bass if and only if it is either a hereditary or an Eichler order of period two.