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College Algebra Exam Review 294

College Algebra Exam Review 294 - Z ŒxŁ is a UFD In Z...

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304 6. RINGS Proof. Let a 1 ; : : : ; a s be nonzero elements in a unique factorization do- main R . Let f 1 ; : : : ; f N be a collection of pairwise nonassociate irre- ducible elements such that each irreducible factor of each a i is an asso- ciate of some f j . Thus each a i has a unique expression of the form a i D u i Q j f n j .a i / j , where u i is a unit. For each j , let m.j / D min i f n j .a i / g . Put d D Q j f m.j / j . I claim that d is a greatest common divisor of f a 1 ; : : : ; a s g . Clearly, d is a common divisor of f a 1 ; : : : ; a s g . Let e be a common divisor of f a 1 ; : : : ; a s g . According to Lemma 6.6.1 , e has the form e D u Q j f k.j / j , where u is a unit and k.j / n j .a i / for all i and j . Hence for each j , k.j / m.j / . Consequently, e divides d . n We say that a 1 ; : : : ; a s are relatively prime if 1 is a greatest common divisor of f a 1 ; : : : ; a s g , that is, if a 1 ; : : : ; a s have no common irreducible factors. Remark 6.6.3. In a principal ideal domain R , a greatest common divisor of two elements a and b is always an element of the ideal aR C bR . But in an arbitrary unique factorization domain R , a greatest common divisor of two elements a and b is not necessarily contained in the ideal
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Unformatted text preview: Z ŒxŁ is a UFD. In Z ŒxŁ , 1 is a greatest common divisor of 2 and x , but 1 62 2 Z ŒxŁ C x Z ŒxŁ . Lemma 6.6.4. In a unique factorization domain, every irreducible is prime. Proof. Suppose an irreducible p in the unique factorization R divides a product ab . If b is a unit, then p divides a . So we can assume that neither a nor b is a unit. If g 1 ²²² g ` and h 1 ²²² h m are irreducible factorizations of a and b , respectively, then g 1 ²²² g ` h 1 ²²² h m is an irreducible factorization of ab . Since p is an irreducible factor of ab , by Lemma 6.6.1 p is an associate of one of the g i ’s or of one of the h j ’s. Thus p divides a or b . n Corollary 6.6.5. Let R be a unique factorization domain domain. Con-sider the following properties of an nonzero, nonunit element p of R : ³ pR is a maximal ideal. ³ pR is a prime ideal. ³ p is prime. ³ p is irreducible....
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