Criteria for a ring to have a left Noetherian left quotient ring

TL;DR

This paper establishes two criteria for rings to possess a left Noetherian left quotient ring, resolving an open problem from the 1970s, and shows such rings have finitely many maximal left denominator sets.

Contribution

It provides the first definitive criteria for identifying rings with a left Noetherian left quotient ring and proves finiteness of maximal left denominator sets in these rings.

Findings

Two criteria for left Noetherian left quotient rings are established.

Rings with such quotient rings have finitely many maximal left denominator sets.

The problem has been open since the 1970s and is now resolved.

Abstract

Two criteria are given for a ring to have a left Noetherian left quotient ring (this was an open problem since 70's). It is proved that each such ring has only finitely many maximal left denominator sets.