(c) Let a E R[r 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+0x² +0x* + • • · , and that multiplication in R[x] is commutative. [Hint. If ab= 1RLi, equate coefficients and solve for bo,b1,b2, . in turn.]

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
100%
please send handwritten solution for part c
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.]
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 1 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,