A generalization of quantales with applications to modules and rings

TL;DR

This paper generalizes quantales through a new lattice structure, develops a point-free approach, and applies these concepts to modules, showing that radical ideals form a frame in commutative rings.

Contribution

It introduces a novel lattice generalization of quantales and extends the theory to modules, providing new insights into the structure of radical ideals.

Findings

The new lattice structure generalizes meet-continuous lattices and quantales.

The point-free approach offers a new perspective on these lattices.

Radical ideals in commutative rings form a frame under the new framework.

Abstract

We introduce a lattice structure as a generalization of meet-continuous lattices and quantales. We develop a point-free approach to these new lattices and apply these results to $R$-modules. In particular, we give the module counterpart of the well known result that in a commutative ring the set of semiprime ideals, that is, the set of radical ideals is a frame.