Near prequantales and applications to star and semistar operations and ideal and module systems

TL;DR

This paper extends the theory of quantales and prequantales to provide a unified, noncommutative framework for various ideal and module systems, offering new representation theorems and constructions.

Contribution

It introduces a generalized noncommutative setting for ideal theoretic operations, connecting and motivating new results in quantale theory and ideal systems.

Findings

Representation theorems for precoherent prequantales

Characterizations of simple commutative quantales

Construction methods for nuclei and semistar operations

Abstract

We show that a generalization of the theory of quantales and prequantales provides a noncommutative and nonassociative abstract ideal theoretic setting for the theories of star operations, semistar operations, semiprime operations, ideal systems, and module systems, and conversely the latter theories motivate new results in the theory of quantales and prequantales. Applications include representation theorems for precoherent prequantales and multiplicative semilattices; characterizations of simple commutative quantales and simple multiplicative lattices; a construction of the largest finitary nucleus smaller than a given nucleus on a precoherent prequantale; a construction of the smallest semistar operation extending a given star operation on an integral domain; and two potentially useful definitions of tight closure for non-Noetherian commutative rings of prime characteristic.