Suppose \(\Gamma \) is a submonoid of a lattice, not containing a line. In this note, we use the natural \(\Gamma \) -grading on the monoid algebra \(R[\Gamma ]\) to prove structural results about the relative K-theory \(K(R[\Gamma ], R)\) . When R contains a field, we prove a decomposition indexed by the rays in \(\Gamma \) and a compatible action by the Witt vectors of R for each \(\mathbb {N}\) -grading of \(\Gamma \) . In characteristic zero, there is additionally an action by Witt vectors for the truncation set \(\Gamma \) . Finally, we apply this to get a ray-like description of \(K_*(R[x_1,\ldots ,x_n])\) proposed by J. Davis.