  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.
