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please send handwritten solution for part a

please send handwritten solution for part a

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 21E: 21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in...
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please send handwritten solution for part a
To submit Let R[x] be the set of all expressions a = do +ajx+azx +..= i=0 where a¡ E R for all nonnegative integers i. Informally, an element of R[x] is like a polynomial except that it can have infinitely many terms. (a) Carefully write down definitions of addition and multiplication operations for R[r], analogous to the definitions for R[x] in the notes. Given a,b E R[x], your defi- nitions should indicate what each coefficient of the sum a+b and product ab is. (b) Let f = ao+ajx+…+a„x" be a polynomial. I can treat f as an element of R[x] by defining an+1,ɑn+2;+…· all to equal 0. This shows that R[x] CR[r]. If you had already proved that R[x] was a ring, how could you use this fact to help you prove RÊ] is a ring? (c) Let a E R[x]] with ao # 0. Prove that a has a multiplicative inverse in R[[x]]: You may assume that the multiplicative identity element in R[r] is 1RL] =1+0x+Ox +0x* + • · · , and that multiplication in R[x] is commutative. [Hint. If ab = 1RL], equate coefficients and solve for bo,b1,b2,-. in turn.]
Transcribed Image Text:To submit Let R[x] be the set of all expressions a = do +ajx+azx +..= i=0 where a¡ E R for all nonnegative integers i. Informally, an element of R[x] is like a polynomial except that it can have infinitely many terms. (a) Carefully write down definitions of addition and multiplication operations for R[r], analogous to the definitions for R[x] in the notes. Given a,b E R[x], your defi- nitions should indicate what each coefficient of the sum a+b and product ab is. (b) Let f = ao+ajx+…+a„x" be a polynomial. I can treat f as an element of R[x] by defining an+1,ɑn+2;+…· all to equal 0. This shows that R[x] CR[r]. If you had already proved that R[x] was a ring, how could you use this fact to help you prove RÊ] is a ring? (c) Let a E R[x]] with ao # 0. Prove that a has a multiplicative inverse in R[[x]]: You may assume that the multiplicative identity element in R[r] is 1RL] =1+0x+Ox +0x* + • · · , and that multiplication in R[x] is commutative. [Hint. If ab = 1RL], equate coefficients and solve for bo,b1,b2,-. in turn.]
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