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hw10 - α R → Z 2 Z and β R → Q Prove that there...

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Math 4124 Monday, April 25 Tenth Homework Due 2:30 p.m., Monday May 2 1. Let R be a commutative ring with a 1 = 0, and let S denote the set of nonzerodivisors 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 with a 1. Suppose there exist ring epimorphisms
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Unformatted text preview: α : R → Z / 2 Z and β : R → Q . Prove that there exists a ring epimorphism θ : R → Z / 2 Z × Q . (2 points) 3. Let R be an integral domain, let S be a multiplicatively closed subset of R which contains 1 but not 0, and let p be a prime of R . Prove that p / 1 is either a prime or a unit of S-1 R . (3 points) 4. Exercise 9.4.7 on p. 311 (1 point) (4 problems, 9 points altogether)...
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