Excerpted from Beachy/Blair, Abstract Algebra, 2nd Ed. © 1996

## § 9.1 Principal Ideal Domains

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

(i) (ab) (b) for all nonzero a,b in D, and

(ii) for any nonzero elements a,b in D there exist q,r in D such that a = bq + r,
where either r = 0 or (r) < (b).

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

(i) d | a and d | b, and

(ii) if c | a and c | b, for c in R, then c | d.

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

(i) p is not a unit of R, and

(ii) if p = ab for a,b in R, then either a or b is a unit of R.

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

(i) each nonzero element a of D that is not a unit can be expressed as a product of irreducible elements of D, and

(ii) in any two such factorizations a = p1 p2 · · · pn = q1 q2 · · · qm the integers n and m are equal and it is possible to rearrange the factors so that qi is an associate of pi, for 1 i n.

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.