Consider the ring R = 3Z and the ideal I = 12Z of R. (a) List explicitly all the cosets of I in R and their elements. (b) Does the factor ring R/I have an identity? If so, which element is the identity? (c) List all the zero-divisors and all the units of R/I. For the units, list their corresponding inverses.
Consider the ring R = 3Z and the ideal I = 12Z of R. (a) List explicitly all the cosets of I in R and their elements. (b) Does the factor ring R/I have an identity? If so, which element is the identity? (c) List all the zero-divisors and all the units of R/I. For the units, list their corresponding inverses.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.1: Ideals And Quotient Rings
Problem 4E: Exercises
If and are two ideals of the ring , prove that is an ideal of .
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Transcribed Image Text:Consider the ring R = 3Z and the ideal I = 12Z of R.
Transcribed Image Text:(a) List explicitly all the cosets of I in R and their elements.
(b) Does the factor ring R/I have an identity? If so, which element is the identity?
(c) List all the zero-divisors and all the units of R/I. For the units, list their
corresponding inverses.
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