3.3. FACTORIZATION IN COMMUTATIVE RINGS 65 3.3 Factorization in Commutative Rings Let R be a commutative ring with identity. Def. • a 2 R − f0g divides b 2 R (notation: a j b) if ax = b for some x 2 R. • a; b 2 R − f0g are associates if a j b and b j a. Def. An element c 2 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 j ab ) p j a or p j 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. p p 3. In Z[ 10] = fa + b 10 j a; b 2 Zg, 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 2 R. 1. a j b iff (b) ⊆ (a). 2. a and b are associates iff (a) = (b), iff a = br for a unit r 2 R. 3. u is a unit iff (u) = R, iff u j r for all r 2 R. (proof) Thm 3.23. Let p; c 2 Rnf0g. 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 2 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 2 N . [ ∗ Proof. The ideal (ai) = (b), where b 2 (an) for certain n 2 N . Then (b) ⊆ (an) ⊆ ∗ i2N (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 2 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 2 S. Then b1 = a2b2 where a2 and b2 are nonzero nonunit, whence a = a1a2b2. Repeating the process, we get b1 j a and bi+1 j bi for i 2 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 2 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 ' : R − f0g ! N such that: 1. '(a) ≤ '(ab) for a; b 2 R − f0g; 2. For a 2 R and b 2 R − f0g, there exist q; r 2 R such that a = qb + r, where either r = 0 or '(r) < '(b). Ex. The following rings are Euclidean domains: 1. The ring Z with '(x) = jxj. 2. A field F with '(x) = 1 for x 2 F − f0g. 3. The ring of polynomials F [x] over a field F with '(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 2 I−f0g such that '(a) = minf'(x) j x 2 I−f0gg: Every b 2 I can be written as b = aq + r with r = 0 or '(r) < '(a). The latter is impossible by r = b − aq 2 I and the minimality of '(a). Therefore r = 0 and b 2 (a). Thus I = (a). So R is a PID. ........................................................................................ From now on, let R be a commutative ring with unity. Def. An element d 2 R is a greatest common divisor (gcd) of a nonempty set X of R if: 1. d j a for all a 2 X; 2. if c j a for all a 2 X, then c j d. Elements x1; ··· ; xn 2 R are relatively prime if 1R is a gcd of fx1; ··· ; xng. If d is a gcd of X, then every associate of d is a gcd of X. Thm 3.28. Let a1; ··· ; an 2 R. 1. If d 2 R is a gcd of fa1; ··· ; ang and d = r1a1 + ··· + rnan for some r1; ··· ; rn 2 R, then (d) = (a1) + ··· + (an); 2. if R is a PID, then a gcd of fa1; ··· ; ang exists and each is of the form r1a1 + ··· + rnan (ri 2 R); 68 CHAPTER 3. RINGS 3. if R is a UFD, then a gcd of fa1; ··· ; ang 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 = minfm1j; m2j; ··· ; mnjg. Show that d = c1 ··· ct is a gcd of fa1; ··· ; ang..