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1, Problem 6. Thus Z is a PID. Now suppose that A is a nonempty subset of the PID R. 7). We show that if d is a greatest common divisor of A, then d generates A, and conversely. 1 Proposition Let R be a PID, with A a nonempty subset of R. Then d is a greatest common divisor of A if and only if d is a generator of A . Proof. Let d be a gcd of A, and assume that A = b . Then d divides every a ∈ A, so d divides all finite sums ri ai . In particular d divides b, hence b ⊆ d ; that is, A ⊆ d . But if a ∈ A then a ∈ b , so b divides a.

6. UNIQUE FACTORIZATION 21 The distinction between irreducible and prime elements disappears in the presence of unique factorization. 3 Definition A unique factorization domain (UFD) is an integral domain R satisfying the following properties: (UF1) Every nonzero element a in R can be expressed as a = up1 . . pn , where u is a unit and the pi are irreducible. (UF2): If a has another factorization, say a = vq1 . . qm , where v is a unit and the qi are irreducible, then n = m and, after reordering if necessary, pi and qi are associates for each i.

Dr = vb1 . . bs wc1 . . ct where u, v and w are units and the di , bi and ci are irreducible. By uniqueness of factorization, a, which is irreducible, must be an associate of some bi or ci . Thus a divides b or a divides c. 5 Definitions and Comments Let A be a nonempty subset of R, with 0 ∈ / A. The element d is a greatest common divisor (gcd) of A if d divides each a in A, and whenever e divides each a in A, we have e|d. If d is another gcd of A, we have d|d and d |d, so that d and d are associates.

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