Similarity of matrices over local rings of length two

Abstract

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 Mn(R) by interpreting them in terms of extensions of R[t]-modules. Using this theory, we describe the similarity classes in Mn(R) for n≤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 Mn(R) turn out to have non-negative integer coefficients.