Invertible module

Summary

An invertible module over a commutative ring $R$ is a module $M$ that has an inverse for the tensor product: some module $N$ with $M \otimes_R N \cong R$. The invertible modules are exactly the finitely generated projective modules of constant rank $1$. Their isomorphism classes form the Picard group of $R$. They are the algebraic form of line bundles, and of invertible sheaves on the affine scheme $\mathrm{Spec}\, R$.