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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
I1 I2 I3 · · · ,there is a subscript m such that In = Im for all n > m.
Theorem 9.1.12. Any principal ideal domain is a unique factorization domain.
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