  Let $R$ be a local principal ideal ring of length two, for example, the ring
$R=\Z/p^2\Z$ with $p$ prime. In this paper we develop a theory of normal forms
for similarity classes in the matrix rings $M_n(R)$ by interpreting them in
terms of extensions of $R[t]$-modules. Using this theory, we describe the
similarity classes in $M_n(R)$ for $n\leq 4$, along with their centralizers.
Among these, we characterize those classes which are similar to their
transposes. Non-self-transpose classes are shown to exist for all $n>3$. When
$R$ has finite residue field of order $q$, we enumerate the similarity classes
and the cardinalities of their centralizers as polynomials in $q$.
Surprisingly, the polynomials representing the number of similarity classes in
$M_n(R)$ turn out to have non-negative integer coefficients.
