7. Let R be a commutative ring and let F = F(R, R) be the set of functions f : R R. Functionsin 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] → Fgiven byp(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 2MATH 4107: Homework 10(c) Give an example showing that p is in general not injective (hint: try R= Z/2Z). Findker(p) when R = Z/pZ where p is prime. Note that the fact that p is not injective meansthat one cannot in general view polynomials in R[X] as R-valued functions on R, whichis in sharp contrast to the case of Z[X], R[X], or C[X].

let R be a commutative ring and F=FR,R be the set of function f:R→R defined as f+gx=fx+gxfgx=fxgx…

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.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.1: Polynomials Over A Ring

Problem 3E

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Question

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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