Perfect Symmetric Rings of Quotients

TL;DR

This paper introduces the concept of perfect symmetric rings of quotients, combining properties of perfect Gabriel filters and symmetric rings of quotients, and extends classical constructions to this new framework.

Contribution

It defines perfect symmetric rings of quotients and Gabriel filters, and adapts existing constructions to develop the total symmetric ring of quotients.

Findings

Established the properties of perfect symmetric rings of quotients

Extended the total right ring of quotients to the symmetric case

Demonstrated the existence of the largest perfect symmetric ring of quotients

Abstract

Perfect Gabriel filters of right ideals and their corresponding right rings of quotients have the desirable feature that every module of quotients is determined solely by the right ring of quotients. On the other hand, symmetric rings of quotients have a symmetry that mimics the commutative case. In this paper, we study rings of quotients that combine these two desirable properties. We define the symmetric versions of a right perfect ring of quotients and a right perfect Gabriel filter -- the perfect symmetric ring of quotients and the perfect symmetric Gabriel filter and study their properties. Then we prove that the standard construction of the total right ring of quotients can be adapted to the construction of the largest perfect symmetric ring of quotients -- the total symmetric ring of quotients. We also demonstrate that Morita's construction of the total right ring of quotients can be adapted to the construction of the total symmetric ring of quotients.