On ideals in quantales -- I

TL;DR

This paper develops a foundational theory of ideals in commutative quantales, exploring properties like primality and decomposition, inspired by ring theory, and sets directions for future research.

Contribution

It introduces a comprehensive framework for ideals in quantales, including prime, radical, primary, and irreducible ideals, and addresses the primary decomposition problem.

Findings

Characterization of prime and semiprime ideals in quantales

Analysis of radical and primary ideals in quantales

Discussion of primary decomposition in the quantale setting

Abstract

Taking a ring-theoretic perspective as our motivation, the main aim of this series is to establish a comprehensive theory of ideals in commutative quantales with an identity element. This particular article focuses on an examination of several key properties related to ideals in quantale, including prime, semiprime, radical, primary, irreducible, and strongly irreducible ideals. Furthermore, we investigate the primary decomposition problem for quantale ideals. In conclusion, we present a set of future directions for further exploration, serving as a natural continuation of this article.