  Let $k$ be an algebraically closed field of characteristic 2. We prove that
the restricted nilpotent commuting variety ${\mathcal C}$, that is the set of
pairs of $(n\times n)$-matrices $(A,B)$ such that $A^2=B^2=[A,B]=0$, is
equidimensional. ${\mathcal C}$ can be identified with the `variety of
$n$-dimensional modules' for ${\mathbb Z}/2{\mathbb Z}\times{\mathbb
Z}/2{\mathbb Z}$, or equivalently, for $k[X,Y]/(X^2,Y^2)$. On the other hand,
we provide an example showing that the restricted nilpotent commuting variety
is not equidimensional for fields of characteristic $>2$. We also prove that if
$e^2=0$ then the set of elements of the centralizer of $e$ whose square is zero
is equidimensional. Finally, we express each irreducible component of
${\mathcal C}$ as a direct sum of indecomposable components of varieties of
${\mathbb Z}/{2{\mathbb Z}}\times{\mathbb Z}/2{\mathbb Z}$-modules.

Rings and Algebras;Representation Theory