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Let R = Z/I, where I consisting of all integral multiples of 10. Recall that the elements of R are cosets of the form r+ I = {r+ 10n : n E Z}, where r E Z. 10Z {0, 10, – 10, 20, -20,...} is the ideal || (a) Explain, in your own words, why 7+ I and 107 + I are the same set of integers. (b) What is the smallest non-negative element of 332 + I ? (c) What is the largest negative element of 332 + I?

Let R = Z/I, where I consisting of all integral multiples of 10. Recall that the elements of R are cosets of the form r+ I = {r+ 10n : n E Z}, where r E Z. 10Z {0, 10, – 10, 20, -20,...} is the ideal || (a) Explain, in your own words, why 7+ I and 107 + I are the same set of integers. (b) What is the smallest non-negative element of 332 + I ? (c) What is the largest negative element of 332 + I?

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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Let R = Z/I, where I = 10Z = consisting of all integral multiples of 10. Recall that the elements of R are cosets of the form r + I = {r+ 10n : n E Z}, where r E Z. {0, 10, – 10, 20, – 20,..} is the ideal (a) Explain, in your own words, why 7+ I and 107 + I are the same set of integers. (b) What is the smallest non-negative element of 332+ I? (c) What is the largest negative element of 332 + I? (d) Which coset is obtained if we multiply 332+I by 5+I, and what is the smallest non-negative element of this coset? (e) Is 5 + I a unit in R? Justify your answer. (f) Give an example of a unit in R. Briefly justify your answer. (g) Is R a field? Briefly justify your answer.
Transcribed Image Text:Let R = Z/I, where I = 10Z = consisting of all integral multiples of 10. Recall that the elements of R are cosets of the form r + I = {r+ 10n : n E Z}, where r E Z. {0, 10, – 10, 20, – 20,..} is the ideal (a) Explain, in your own words, why 7+ I and 107 + I are the same set of integers. (b) What is the smallest non-negative element of 332+ I? (c) What is the largest negative element of 332 + I? (d) Which coset is obtained if we multiply 332+I by 5+I, and what is the smallest non-negative element of this coset? (e) Is 5 + I a unit in R? Justify your answer. (f) Give an example of a unit in R. Briefly justify your answer. (g) Is R a field? Briefly justify your answer.
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