Sum-Check as an Algebraic Tensor Reduction: Part II

In this part of our series, we start introducing the algebraic language needed to formalize sum-check as a tensor reduction. We start with the basics of rings and modules. Rings generalize fields by dropping the requirement that every non-zero element has a multiplicative inverse. Modules then generalize vector spaces by allowing scalars to come from a ring instead of a field. In this post, we’ll use plenty of examples to make these ideas concrete and build intuition along the way.