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We introduce several commutative rings, the snake rings, that have strongconnections to cluster algebras. The elements of these rings are residueclasses of unions of certain labeled graphs that were used to constructcanonical bases in the theory of cluster algebras. We obtain several rings byvarying the conditions on the structure as well as the labelling of the graphs.The most restrictive form of this ring is isomorphic to the ring$\mathbb{Z}[x,y]$ of polynomials in two variables over the integers. A moregeneral form contains all cluster algebras of unpunctured surface type.  The definition of the rings requires the snake graph calculus which iscompleted in this paper building on two earlier articles on the subject.Identities in the snake ring correspond to bijections between the posets ofperfect matchings of the graphs. One of the main results of the current paperis the completion of the explicit construction of these bijections.

Snake graph calculus and cluster algebras from surfaces III: Band graphs and snake rings