Additive closed symmetric monoidal structures on R-modules

TL;DR

This paper classifies additive closed symmetric monoidal structures on categories of R-modules, revealing the diversity and constraints of such structures depending on the ring R, and providing structural insights.

Contribution

It provides a complete classification of these structures using Watts' theorem and explores their existence and uniqueness for various rings R.

Findings

Some rings admit no such structures.

Some rings admit exactly one such structure.

Some rings admit a proper class of such structures.

Abstract

In this paper, we classify additive closed symmetric monoidal structures on the category of left R-modules by using Watts' theorem. An additive closed symmetric monoidal structure is equivalent to an R-module Lambda_{A,B} equipped with two commuting right R-module structures represented by the symbols A and B, an R-module K to serve as the unit, and certain isomorphisms. We use this result to look at simple cases. We find rings R for which there are no additive closed symmetric monoidal structures on R-modules, for which there is exactly one (up to isomorphism), for which there are exactly seven, and for which there are a proper class of isomorphism classes of such structures. We also prove some general structual results; for example, we prove that the unit K must always be a finitely generated R-module.