Additive closed symmetric monoidal structures on R-modules

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 ΛA,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 structural results; for example, we prove that the unit K must always be a finitely generated R-module.