Fusion rings with few degrees

The main result characterizes fusion rings whose degree set consists of 1 and a fixed prime p. A well known theorem of Isaacs and Passman on finite groups whose irreducible character degrees are 1 and p follows as a corollary. Numerically compatible (NC) fusion rings are defined as a potential generalization of the Grothendieck rings of fusion categories; and the NC fusion rings of order p  , p2p2, p3p3, pq  , and pq2pq2 (where p, q are primes) are classified in terms of certain standard table algebras, as another application of the main theorem.