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College Algebra Exam Review 292

# College Algebra Exam Review 292 - b s Show that p divides...

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302 6. RINGS 6.5.15. Let R be a principal ideal domain. Show that any pair of nonzero elements a;b 2 R have a greatest common divisor and that for any greatest common divisor d , we have d 2 aR C bR . Show that a and b are relatively prime if, and only if, 1 2 aR C bR . 6.5.16. Let a be an irreducible element of a principal ideal domain R . If b 2 R and a does not divide b , show that a and b are relatively prime. 6.5.17. Suppose a and b are relatively prime elements in a principal ideal domain R . Show that aR \ bR D abR and aR C bR D R . Show that R=abR Š R=aR ˚ R=bR . This is a generalization of the Chinese re- mainder theorem, Exercise 6.3.11 . 6.5.18. Consider the ring of polynomials in two variables over any ﬁeld R D KŒx;yŁ . (a) Show that the elements x and y are relatively prime. (b) Show that it is not possible to write 1 D p.x;y/x C q.x;y/y , with p;q 2 R . (c) Show that R is not a principal ideal domain, hence not a Eu- clidean domain. 6.5.19. Show that a prime element in any integral domain is irreducible. 6.5.20. Let p be a prime element in an integral domain and suppose that p divides a product b 1 b 2 ±±±
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Unformatted text preview: b s . Show that p divides one of the b i . 6.5.21. Let aR be an ideal in a principal ideal domain R . Show that R=aR is a ring with only ﬁnitely many ideals. 6.5.22. (a) Let a be an element of an integral domain R . Show that R=aR is an integral domain if, and only if, a is prime. (b) Let a be an element of a principal ideal domain R . Show that R=aR is a a ﬁeld if, and only if, a is irreducible. (c) Show that a quotient R=I of a principal ideal domain R by a nonzero proper ideal I is an integral domain if, and only if, it is a ﬁeld. (d) Show that an ideal in a principal ideal domain is maximal if, and only if, it is prime. 6.5.23. Consider the ring of formal power series KŒŒxŁŁ with coefﬁcients in a ﬁeld K . (a) Show that the units of KŒŒxŁŁ are the power series with nonzero constant term (i.e., elements of the form ˛ C xf , where ˛ ¤ , and f 2 KŒŒxŁŁ )....
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