Right Mori Orders

TL;DR

This paper introduces right Mori orders, a class of prime Goldie rings with specific ideal chain conditions, and explores their structural properties and ideal decompositions.

Contribution

It establishes that right Mori orders are closed under Morita-equivalence and characterizes their regular right divisorial ideals.

Findings

Right Mori orders are closed under Morita-equivalence.

Regular right divisorial ideals are contained in finitely many prime ideals.

Such ideals can be expressed as finite intersections of $ u$-irreducible ideals.

Abstract

In this paper we study \textit{right Mori Orders}, which are those prime Goldie rings that satisfy the ascending chain condition on regular integral right divisorial right ideals. We will show that the class of right Mori orders is closed with respect to Morita-equivalence. We also prove that each regular right divisorial right ideal of a right Mori order is contained in only finitely many right divisorial completely prime right ideals. Moreover, we show that such right divisorial ideals can be represent as a finite intersection of $\nu$-irreducible ideals of the form $aS:_rb$ for some regular $a,b\in S$.