The Enumeration of Finite Rings

Let p be a fixed prime. We show that the number of isomorphism classes of finite rings of order p^n is p^\alpha, where \alpha=(4/27)n^3+O(n^{5/2}). This result was stated (with a weaker error term) by Kruse and Price in 1969; a problem with their proof was pointed out by Knopfmacher in 1973. We also show that the number of isomorphism classes of finite commutative rings of order p^n is p^\beta, where \beta=(2/27)n^3+O(n^{5/2}). This result was stated (again with a weaker error term) by Poonen in 2008, with a proof that relies on the problematic step in Kruse and Price's argument.