  In this paper, we explain how the abstract notion of a differential bundle in
a tangent category provides a new way of thinking about the category of modules
over a commutative ring and its opposite category. MacAdam previously showed
that differential bundles in the tangent category of smooth manifolds are
precisely smooth vector bundles. Here we provide characterizations of
differential bundles in the tangent categories of commutative rings and
(affine) schemes. For commutative rings, the category of differential bundles
over a commutative ring is equivalent to the category of modules over that
ring. For affine schemes, the category of differential bundles over the Spec of
a commutative ring is equivalent to the opposite category of modules over said
ring. Finally, for schemes, the category of differential bundles over a scheme
is equivalent to the opposite category of quasi-coherent sheaves of modules
over that scheme.
