article

Title: Similarity of matrices over local rings of length two

Authors: Amritanshu Prasad, Pooja Singla and Steven Spallone

Issue: Volume 64 (2015), Issue 2, 471-514

Abstract:

$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.$