Primitive quantales

TL;DR

This paper extends the concept of primitive rings to primitive quantales, establishing their properties, relationships with prime and field quantales, and proving a density theorem for strongly primitive quantales.

Contribution

It introduces primitive quantales as a generalization of primitive rings, explores their properties, and proves new theorems including a density theorem.

Findings

Primitive quantales are constructed from primitive rings.

Primitive quantales are prime.

Strongly primitive commutative quantales are field quantales.

Abstract

We generalize Jacobson's notion of primitive ring to the setting of quantales. We show that every primitive ring gives rise to a primitive quantale of ideals. We then prove a density theorem for strongly primitive quantales. Furthermore, we show that primitive quantales are prime and commutative strongly primitive quantales are field quantales.