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 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.

💡 Research Summary

The paper undertakes a complete classification of additive closed symmetric monoidal structures on the category of left modules over a ring R. An “additive closed symmetric monoidal structure” on Mod‑R is one whose tensor product is additive in each variable, admits an internal Hom, and whose symmetry isomorphism satisfies the usual coherence conditions. The authors show that any such structure can be described entirely by a single R‑bimodule Λ equipped with two commuting right R‑actions, denoted A and B, together with a distinguished R‑module K that plays the role of the unit object.

The key technical tool is Watts’ theorem, which asserts that any additive right exact functor from Mod‑R to itself is naturally isomorphic to tensoring with a fixed R‑module. By applying this theorem to the two functors “‑⊗ X” and “X⊗ ‑” arising from a putative monoidal product, the authors deduce that the product must have the form
\