Similarity of matrices over local rings of length two Amritanshu PrasadPooja SinglaSteven Spallone 15A2105E1520G25similarity classesmatriceslocal ringsextensions Let $R$ be a (commutative) local principal ideal ring of length two, for example, the ring $R=\mathbb{Z}/p^2\mathbb{Z}$ 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 $n\leq4$, 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. Indiana University Mathematics Journal 2015 text pdf 10.1512/iumj.2015.64.5500 10.1512/iumj.2015.64.5500 en Indiana Univ. Math. J. 64 (2015) 471 - 514 state-of-the-art mathematics http://iumj.org/access/