We introduce several commutative rings, the snake rings, that have strong connections to cluster algebras and that are interesting on their own right as combinatorially defined rings. The elements of these rings are residue classes of unions of certain labeled graphs that were instrumental to construct canonical bases in the theory of cluster algebras. We obtain several rings by varying the conditions on the structure as well as the labeling of the graphs. A general form of this ring contains all cluster algebras of unpunctured surface type. The definition of the rings requires the snake graph calculus which we also complete in this article building on two earlier articles on the subject. Identities in the snake ring correspond to bijections between the posets of perfect matchings of the graphs. One of the main results of this article is the completion of the explicit construction of these bijections.