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Definition 9.1.1.
An integral domain D is called a
Euclidean domain
if for each nonzero element x in D
there is assigned a nonnegative integer
(x) such that
Theorem 9.1.2.
Any Euclidean domain is a principal ideal domain.
Let a and b be elements of a commutative ring R with 1.
Then a is called an
associate
of b if a = bu for some unit u in R.
Definition 9.1.4.
Let a and b be elements of a commutative ring R with identity.
An element d of R is called a
greatest common divisor
of a and b if
Proposition 9.1.6.
Let D be a principal ideal domain.
If a and b are nonzero elements of D,
then D contains a greatest common divisor of a and b,
of the form as+bt for s,t in D.
Furthermore, any two greatest common divisors of a and b are associates.
In the situation of the above proposition,
in an integral domain D, we say that a and b are
relatively prime
if aD + bD = D.
Definition 9.1.7.
Let R be a commutative ring with identity.
A nonzero element p of R is said to be
irreducible
if
Proposition 9.1.8.
Let p be an irreducible element of the principal ideal domain D.
If a,b are in D and p|ab, then either p|a or p|b.
Proposition 9.1.9.
Let D be a principal ideal domain,
and let p be a nonzero element of D.
Then p is irreducible in D if and only if
pD is a prime ideal of D.
Definition 9.1.10.
Let D be an integral domain. Then D is called a
unique factorization domain
if
Lemma 9.1.11.
Let D be a principal ideal domain.
In any collection of ideals
I_{1} I_{2} I_{3} · · · ,
there is a subscript m such that I_{n} = I_{m} for all n > m.
Theorem 9.1.12.
Any principal ideal domain is a unique factorization domain.