The enumeration of finite rings
Simon R. Blackburn⋆, K. Robin McLean · Journal of the London Mathematical Society · 2022
Let p $p$ be a fixed prime. We show that the number of isomorphism classes of finite rings of order p n $p^n$ is p α $p^\alpha$ , where α = 4 27 n 3 + O ( n 5 / 2 ) $\alpha =\frac{4}{27}n^3+\mathnormal {O}(n^{5/2})$ . 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 $p^n$ is p β $p^\beta$ , where β = 2 27 n 3 + O ( n 5 / 2 ) $\beta =\frac{2}{27}n^3+\mathnormal {O}(n^{5/2})$ . 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.