The enumeration of finite rings

TL;DR

This paper precisely counts the number of finite rings and finite commutative rings of order p^n, providing exact formulas with error bounds, correcting earlier incomplete proofs from past decades.

Contribution

It establishes exact asymptotic formulas for the number of finite rings and commutative rings of order p^n, correcting previous flawed proofs and refining earlier estimates.

Findings

Number of finite rings of order p^n is p^{(4/27)n^3 + O(n^{5/2})}

Number of finite commutative rings of order p^n is p^{(2/27)n^3 + O(n^{5/2})}

Provides rigorous proof and correction of earlier conjectures and partial results.

Abstract

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 $\alpha=\frac{4}{27}n^3+O(n^{5/2})$. 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 $\beta=\frac{2}{27}n^3+O(n^{5/2})$. 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.