7. Let R be a commutative ring and let F = F(R, R) be the set of functions f: R→ R. Functions in F can be added and multiplied pointwise using the operations from R, i.e. define (f+ 9)(z) = f(x) + g(x) (f9)(z) = f(r)g(x) (a) Prove F is a commutative ring.
7. Let R be a commutative ring and let F = F(R, R) be the set of functions f: R→ R. Functions in F can be added and multiplied pointwise using the operations from R, i.e. define (f+ 9)(z) = f(x) + g(x) (f9)(z) = f(r)g(x) (a) Prove F is a commutative 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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![7. Let R be a commutative ring and let F = F(R, R) be the set of functions f : R R. Functions
in F can be added and multiplied pointwise using the operations from R, i.e. define
(S + 9)(x) = f(x) + g(x)
(f9)(2) = f(a)g(x)
(a) Prove F is a commutative ring.
(b) Let I denote the identity function 1(r) = r in F. Prove that the map p : R[X] → F
given by
p(a, X" +. + a1 X + ao) = a," + . + a1l + ao
...
is a ring homomorphism, where R[X] is the ring of polynomials with coefficients in R.
1 of 2
MATH 4107: Homework 10
(c) Give an example showing that p is in general not injective (hint: try R= Z/2Z). Find
ker(p) when R = Z/pZ where p is prime. Note that the fact that p is not injective means
that one cannot in general view polynomials in R[X] as R-valued functions on R, which
is in sharp contrast to the case of Z[X], R[X], or C[X].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F744af276-3b7c-49cb-a92c-d6b2708fe35d%2F2d2d6b80-6885-4533-a3d8-08884326abec%2Fcdnapy6s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:7. Let R be a commutative ring and let F = F(R, R) be the set of functions f : R R. Functions
in F can be added and multiplied pointwise using the operations from R, i.e. define
(S + 9)(x) = f(x) + g(x)
(f9)(2) = f(a)g(x)
(a) Prove F is a commutative ring.
(b) Let I denote the identity function 1(r) = r in F. Prove that the map p : R[X] → F
given by
p(a, X" +. + a1 X + ao) = a," + . + a1l + ao
...
is a ring homomorphism, where R[X] is the ring of polynomials with coefficients in R.
1 of 2
MATH 4107: Homework 10
(c) Give an example showing that p is in general not injective (hint: try R= Z/2Z). Find
ker(p) when R = Z/pZ where p is prime. Note that the fact that p is not injective means
that one cannot in general view polynomials in R[X] as R-valued functions on R, which
is in sharp contrast to the case of Z[X], R[X], or C[X].
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