SS, +, >, and a field F := F, +, . >. (a) Prove that the integral domain D is a field if and only if each equation ax = b for a, b € D and a 0 has a unique solution in D. (b) Given a ring S. For each s € S define a mapping og : SS by o(a) = s. a for each a € S. If S is a commutative ring, explain why Ker o, ‡ {0} if and only if s = 0 or s is a zero divisor in S. (c) For S a field and s‡0, explain why σ, defined in (b) is an isomorphism of the additive group of S onto itself.

SS, +, >, and a field F := F, +, . >. (a) Prove that the integral domain D is a field if and only if each equation ax = b for a, b € D and a 0 has a unique solution in D. (b) Given a ring S. For each s € S define a mapping og : SS by o(a) = s. a for each a € S. If S is a commutative ring, explain why Ker o, ‡ {0} if and only if s = 0 or s is a zero divisor in S. (c) For S a field and s‡0, explain why σ, defined in (b) is an isomorphism of the additive group of S onto itself.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.3: The Field Of Quotients Of An Integral Domain

Problem 10E: Since this section presents a method for constructing a field of quotients for an arbitrary integral...

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