Exercise 4. (a) Let R be an integral domain where all irreducible elements are prime. Show that if a nonzero element a E

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Exercise 4 A Let R Be An Integral Domain Where All Irreducible Elements Are Prime Show That If A Nonzero Element A E 1 (84.05 KiB) Viewed 20 times

Exercise 4. (a) Let R be an integral domain where all irreducible elements are prime. Show that if a nonzero element a E R admits a factorization into irreducibles, then this factorization is unique.3 (b) Given a unique factorization domain (UFD), namely an integral domain where each nonzero element admits a unique factorization into irreducibles, prove that each irreducible element is prime. Here a B means that 8 is a multiple of a in R, i.e., B = ay for some 7 e R. ?Namely, a = saj . Am, with e invertible and each a; an irreducible element, and m> 0 (possibly m= 0). 3As usual, this means that if a = f'd...om is another factorization, then mrn and there exists a permutation O E Sn such that a = Eido(i) for some invertible element ki (and actually € = £'£1... En). So, strictly speaking, the factorization is unique up to invertibles and up to permutation of the factors.