Let H ={[1 a] : a in Z} {[0 1] Define a map φ : Z → H by φ(n) = [1 n] for all n in Z . [o 1] a. Show that φ is one-to-one. (Definition of 1 − 1: If φ(x) = φ(y), then x = y. To prove this, Assume φ(x) = φ(y) and show that x = y.) b. ) Show that φ is onto. (Definition of onto: For every y ∈ H, there is x ∈ Z such that φ(x) = y. To prove this, start with a matrix in H and then find an element in Z that is mapped to that matrix.) c. Show that φ is operation preserving. (Show: φ(x+y) = φ(x)φ(y). To do this, compute φ(x+y) and then computer φ(x)φ(y) and compare them.) d Is H ≈ Z? Explain.

Let H ={[1 a] : a in Z} {[0 1] Define a map φ : Z → H by φ(n) = [1 n] for all n in Z . [o 1] a. Show that φ is one-to-one. (Definition of 1 − 1: If φ(x) = φ(y), then x = y. To prove this, Assume φ(x) = φ(y) and show that x = y.) b. ) Show that φ is onto. (Definition of onto: For every y ∈ H, there is x ∈ Z such that φ(x) = y. To prove this, start with a matrix in H and then find an element in Z that is mapped to that matrix.) c. Show that φ is operation preserving. (Show: φ(x+y) = φ(x)φ(y). To do this, compute φ(x+y) and then computer φ(x)φ(y) and compare them.) d Is H ≈ Z? Explain.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 1E: Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary...

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