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Unformatted text preview: Math 4124 Monday, April 25 April 25, Ungraded Homework Exercise 7.5.2 on page 264 Let R be an integral domain and let D be a multiplicatively closed subset of R which contains 1 but not 0. Prove that the ring of fractions D- 1 R is isomorphic to a subring of the quotient field of R (hence is also an integral domain). Let S = R- 0 and let : R S- 1 R denote the natural homomorphism defined by ( r ) = r / 1. Note that S- 1 R is the quotient field of the integral domain R , and is a monomorphism. In this situation we usually identify R with ( R ) , but we wont do so here. Since ( d ) 6 = for all d D , we see that ( d ) is invertible for all d D . By the universal property for D- 1 R , we may extend to a map : D- 1 R S- 1 R . Specifically if : R D- 1 R is the natural homomorphism defined by ( r ) = r / 1, then = . All that remains to prove is that is a monomorphism, equivalently ker = 0, so suppose r / d ker where r...
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