  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.
