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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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Given an integral domain D:D, +, >, a ring Question 2. 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 S→ S by os(a) = sa for each a € S. If S is a commutative ring, explain why Ker os {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 o, defined in (b) is an isomorphism of the additive group of S onto itself. (d) Prove that if C denotes any collection of subfields of a field F, then the intersection of all the fields in C is also a subfield of F.
Transcribed Image Text:Given an integral domain D:D, +, >, a ring Question 2. 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 S→ S by os(a) = sa for each a € S. If S is a commutative ring, explain why Ker os {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 o, defined in (b) is an isomorphism of the additive group of S onto itself. (d) Prove that if C denotes any collection of subfields of a field F, then the intersection of all the fields in C is also a subfield of F.
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