3.3 Factorization in Commutative Rings

3.3. FACTORIZATION IN COMMUTATIVE RINGS 65

Let R be a commutative ring with identity.

Def.

• a ∈ R − {0} divides b ∈ R (notation: a | b) if ax = b for some x ∈ R.

• a, b ∈ R − {0} are associates if a | b and b | a.

Def. An element c ∈ R is irreducible if

1. c is a nonzero nonunit;

2. c = ab ⇒ a or b is a unit.

An element p of R is prime if

1. p is a nonzero nonunit;

2. p | ab ⇒ p | a or p | b.

Ex.

1. In Z, every prime number p is both irreducible and prime.

2. In Z6, 2 is prime but not irreducible since 2 · 4 = 2 and neither 2 nor 4 are units. √ √ 3. In Z[ 10] = {a + b 10 | a, b ∈ Z}, 2 is irreducible but not prime (HW 3.3.3)...... Now we assume that R is an integral domain. The divisibility in R can be interpreted in terms of principal ideals.

Thm 3.22. Let a, b, u ∈ R.

1. a | b iﬀ (b) ⊆ (a).

2. a and b are associates iﬀ (a) = (b), iﬀ a = br for a unit r ∈ R.

3. u is a unit iﬀ (u) = R, iﬀ u | r for all r ∈ R.

(proof)

Thm 3.23. Let p, c ∈ R\{0}.

1. p is prime iﬀ (p) is a nonzero prime ideal; 66 CHAPTER 3. RINGS

2. c is irreducible iﬀ (c) is maximal in the set of all proper principal ideals of R;

3. every prime element is irreducible;

4. if R is a PID, then an element is irreducible iﬀ it is prime;

5. every associate of an irreducible [resp. prime] element of R is irreducible [resp. prime].

(proof)

Def. An integral domain R is a unique factorization domain (UFD) if

1. every nonzero nonunit element a of R can be written as a = c1 ··· cn, where c1, ··· , cn are irreducible;

2. if a = d1 ··· dm where d1, ··· , dm are irreducible, then m = n, and for some σ ∈ Sn, ai and bσ(i) are associates. Thm 3.24. In a UFD, an element is irreducible iﬀ it is prime.

(proof) We will introduce Euclidean domain. A Euclidean domain is a principal ideal domain; a principal ideal domain is a UFD. A ring R is a UFD implies that the polynomial ring R[x] is a UFD. The following lemma says that a PID is a Noether ring.

Lem 3.25. Let (a1) ⊆ (a2) ⊆ · · · be a chain of ideals in a principal ideal domain R. Then ∗ (an) = (an+1) = ··· for certain n ∈ N .

[ ∗ Proof. The ideal (ai) = (b), where b ∈ (an) for certain n ∈ N . Then (b) ⊆ (an) ⊆ ∗ i∈N (an+1) ⊆ · · · ⊆ (b). Hence (an) = (an+1) = ··· . Thm 3.26. Every principal ideal domain R is a UFD.

Proof. Let S be the set of all elements of R which cannot be factored as a ﬁnite product of irreducibles. We claim that S = ∅. Suppose on the contrary, a ∈ S. Then a 6= 0 is not irreducible. So a = a1b1 where a1 and b1 are nonzero nonunits, and at least one of a1 and b1 is in S, say b1 ∈ S. Then b1 = a2b2 where a2 and b2 are nonzero nonunit, whence a = a1a2b2. Repeating the process, we get b1 | a and bi+1 | bi for i ∈ N. Then (a) ( (b1) ( (b2) ( ··· , a contradiction to the preceding lemma. Thus S = ∅. If a = c1 ··· cn = d1 ··· dm where ci and dj are irreducible (and thus prime), then c1 divides certain dj so that c1 and dj are associates. By induction, we can show that n = m, and there is σ ∈ Sn such that ai and bσ(i) are associates. 3.3. FACTORIZATION IN COMMUTATIVE RINGS 67

Def. An integral domain R is a Euclidean domain if there is a function

ϕ : R − {0} → N such that:

1. ϕ(a) ≤ ϕ(ab) for a, b ∈ R − {0};

2. For a ∈ R and b ∈ R − {0}, there exist q, r ∈ R such that a = qb + r, where either r = 0 or ϕ(r) < ϕ(b).

Ex. The following rings are Euclidean domains:

1. The ring Z with ϕ(x) = |x|. 2. A ﬁeld F with ϕ(x) = 1 for x ∈ F − {0}.

3. The ring of polynomials F [x] over a ﬁeld F with ϕ(f) = degree of f.

Thm 3.27. Every Euclidean domain R is a principal ideal domain.

Proof. If I is a nonzero ideal of R, choose a ∈ I−{0} such that ϕ(a) = min{ϕ(x) | x ∈ I−{0}}. Every b ∈ I can be written as b = aq + r with r = 0 or ϕ(r) < ϕ(a). The latter is impossible by r = b − aq ∈ I and the minimality of ϕ(a). Therefore r = 0 and b ∈ (a). Thus I = (a). So R is a PID.

...... From now on, let R be a commutative ring with unity.

Def. An element d ∈ R is a greatest common divisor (gcd) of a nonempty set X of R if:

1. d | a for all a ∈ X;

2. if c | a for all a ∈ X, then c | d.

Elements x1, ··· , xn ∈ R are relatively prime if 1R is a gcd of {x1, ··· , xn}. If d is a gcd of X, then every associate of d is a gcd of X.

Thm 3.28. Let a1, ··· , an ∈ R.

1. If d ∈ R is a gcd of {a1, ··· , an} and d = r1a1 + ··· + rnan for some r1, ··· , rn ∈ R, then (d) = (a1) + ··· + (an);

2. if R is a PID, then a gcd of {a1, ··· , an} exists and each is of the form r1a1 + ··· + rnan (ri ∈ R); 68 CHAPTER 3. RINGS

3. if R is a UFD, then a gcd of {a1, ··· , an} exists. Proof.

1. easy.

2. follows from 1.

mi1 mit 3. Factorize ai = c1 ··· ct for i = 1, ··· , n with c1, ··· , ct not associate irreducible k1 kt elements and mij ≥ 0. Let kj = min{m1j, m2j, ··· , mnj}. Show that d = c1 ··· ct is a gcd of {a1, ··· , an}.