Page 114 - Matrices theory and applications

6
Matrices with Entries in a Principal
Ideal Domain; Jordan Reduction

6.1 Rings, Principal Ideal Domains

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 (a, b)R(c, d) ⇔
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.Ifthere 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.

6.1.1 Facts About Principal Ideal Domains

Definition 6.1.1 A commutative ring A is a principal ideal domain if
every ideal in A is principal: For every ideal I there exists a ∈ A such that
I =(a).

A field is a principal ideal domain that has only two ideals, (0) and (1).
The set ZZ of rational integers and the polynomial algebra over a field k,