Picard group of a ring

Summary

The Picard group $\operatorname{Pic}(R)$ of a commutative ring $R$ collects the isomorphism classes of rank one projective modules over $R$. Its group law is the tensor product. It measures how far such modules are from being free. For an integral domain it is the group of invertible ideals modulo principal ones, and for a Dedekind domain it is the ideal class group. It is the Picard group of the affine scheme $\operatorname{Spec} R$.