New criteria for a ring to have a semisimple left quotient ring

TL;DR

This paper introduces four new criteria, based on different approaches, to determine when a ring has a semisimple left quotient ring, extending and simplifying Goldie's Theorem with explicit conditions.

Contribution

The paper presents four novel criteria for semisimple left quotient rings, utilizing maximal left denominator sets, minimal primes, and simplified checks, offering new perspectives and tools beyond Goldie's Theorem.

Findings

Criteria based on maximal left denominator sets and their intersections.

Explicit description of maximal left denominator sets via minimal primes.

Simplified conditions reducing Goldie's Theorem to prime cases.

Abstract

Goldie's Theorem (1960), which is one of the most important results in Ring Theory, is a criterion for a ring to have a semisimple left quotient ring. The aim of the paper is to give four new criteria (using a completely different approach and new ideas). The first one is based on the recent fact that for an arbitrary ring $R$ the set $\CM$ of maximal left denominator sets of $R$ is a non-empty set: Theorem (The First Criterion). A ring $R$ has a semisimple left quotient ring $Q$ iff $\CM $ is a finite set, $\bigcap_{S\in \CM} \ass (S) =0$ and, for each $S\in \CM$, the ring $S^{-1}R$ is a simple left Artinian ring. In this case, $Q\simeq \prod_{S\in \CM} S^{-1}R$. The Second Criterion is given via the minimal primes of $R$ and goes further then the First one %and Goldie's Theorem in the sense that it describes explicitly the maximal left denominator sets $S$ via the minimal primes of $R$. The Third Criterion is close to Goldie's Criterion but it is easier to check in applications (basically, it reduces Goldie's Theorem to the prime case). The Fourth Criterion is given via certain left denominator sets.