Additive closed symmetric monoidal structures on $R$-modules
Mark Hovey
Wesleyan University
mhovey@wesleyan.edu
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.