Matrices with Entries in a Principal Ideal Domain; Jordan Reduction

Abstract

In this Chapter we consider commutative integral domains A (see Chapter 2). In particular, such a ring A can be embeded in its field of fractions, which is the quotient of A × (A \ {0}) by the equivalence relation $\left( {a,b} \right)\mathcal{R}\left( {c,d} \right) \Leftrightarrow$ ad = bc. The embedding is the map a↦(a,1). In a ring A the set of invertible elements is denoted by A^*. If a, b ∈ A are such that b = ua with u ∈ A^*, we say that a and b are associated, and we write a ∼ b, which amounts to saying that aA = bA. If there exists c ∈ A such that ac = b, we say that a divides b and write a|b. Then the quotient c is unique and is denoted by b/a. We say that b is a prime, or irreducible, element if the equality b = ac implies that one of the factors is invertible.

An ideal I in a ring A is an additive subgroup of A such that A · I . I: a ∈ A, x ∈ I imply ax ∈ I. For example, if b ∈ A, the subset bA is an ideal, denoted by (b). Ideals of the form (b) are called principal ideals.