Picard group of a ring ($\mathrm{Pic}(A)$)

The Picard group of a commutative ring $A$ consists of isomorphism classes of finitely generated projective modules that have rank one at every prime ideal. Tensor product is the group law, $[A]$ is the identity, and the module dual supplies the inverse. For an integral domain, every such module can be represented by an invertible fractional ideal after choosing a basis over the fraction field.