  For a Hopf-Galois structure on a Galois extension $L/K$ of fields that arises
from a finite nilpotent $\mathbb{F}_p$-algebra $A$, we look at the Galois
correspondence ratio, which measures the failure of surjectivity of the Galois
correspondence for the Hopf-Galois structure on $L/K$. Using methods of
elementary linear algebra, we observe that the number of subgroups of the
adjoint group of $A$ is equal to the number of subgroups of the additive group
of $N$. Then we count left ideals of $A$ and thereby determine the GCR for all
nilpotent $\mathbb{F}_p$-algebras of dimension 4, and also show that for a set
of $\mathbb{F}_p$-algebras of arbitrary dimension $n$ and exponent $e$, the GCR
approaches 0 for large $p$, $n$ or $e$.
