Let R[r] be the set of all expressions 00 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*] by defining an+1,&n+2;+… all to equal 0. This shows that R[x] CR[x]. If you had already proved that R[x] was a ring, how could you use this fact to help you prove R] is a ring? [
Let R[r] be the set of all expressions 00 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*] by defining an+1,&n+2;+… all to equal 0. This shows that R[x] CR[x]. If you had already proved that R[x] was a ring, how could you use this fact to help you prove R] is a ring? [
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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please send handwritten solution for part b
![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.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf2e7c4c-f19c-4197-9d83-557cde502097%2F0657315f-7809-44d8-9624-61a7c59f14b2%2Fywibd55_processed.jpeg&w=3840&q=75)
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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