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hw8 - Q x is a UFD Let r = f g ∈ Q x(the field of...

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Math 5125 Friday, October 14 Eighth Homework Due 9:05 a.m., Friday October 21 1. Let R be a commutative ring with a 1 = 0, and let S denote the set of non-zerodivisors of R (that is { s R | sr = 0 for all r R - 0 } ). (a) Prove that S is a multiplicatively closed subset of R which contains 1 but not 0. (b) Prove that every element of S - 1 R is either a unit (that is, u such that uv = 1 for some v ) or a zero divisor (zero divisor includes 0). (3 points) 2. Let R be a commutative ring and let S be a multiplicatively closed subset of R which contains 1 but not 0 (and may include zero divisors). Prove or give a counterexample to the following statement: S - 1 R = 0. (2 points) 3. Let R be a commutative ring and let I , J be comaximal ideals (that is, R = I + J ). Prove that R / ( IJ ) = R / I × R / J . (2 points) 4. For this problem we will assume the result that
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Unformatted text preview: Q [ x ] is a UFD. Let r = f / g ∈ Q ( x ) (the field of fractions of Q [ x ] ) with f , g ∈ Q [ x ] \ 0, coprime (no common irreducible factor), and at least one of f , g has degree nonzero. (a) Prove that there is a unique monomorphism θ r : Q ( x ) → Q ( x ) such that θ r ( x ) = r . (b) If f and g both have degree at most 1, prove that θ r is an automorphism. Hint: if r = ax + b cx + d , consider the inverse of the matrix ± a b c d ² and use this to construct s ∈ Q ( x ) such that θ r θ s ( x ) = θ s θ r ( x ) = x . (4 points) 5. Section 8.3, Exercise 2 on page 293. (2 points) (5 problems, 13 points altogether)...
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