Please prove the following with clear steps and/or related theorem.Problem 3.(i) Prove or disprove: if R is a commutative ring and P and Q are polyno-mials in R[a], then P = Q if and only if P(a) = Q(a) for all a e R.(ii) Prove or disprove: if R is a commutative ring and P E R[x] is a polynomial whosecoefficients are zero divisors in R, then P is a zero divisor in R[x].(iii) (Bonus) Prove or disprove: if R is a commutative ring and P is a zero divisor in R[x],then every coefficient in P is a zero divisor in R.

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Answered: Problem 3. (i) Prove or disprove: if R…

Problem 3. (i) Prove or disprove: if R is a commutative ring and P and Q are polyno- mials in R[x], then P = Q if and only if P(a) = Q(a) for all a e R. %3D

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.1: Definition Of A Ring

Problem 54E

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Question

Please prove the following with clear steps and/or related theorem.
Problem 3.
(i) Prove or disprove: if R is a commutative ring and P and Q are polyno-
mials in R[a], then P = Q if and only if P(a) = Q(a) for all a e R.
(ii) Prove or disprove: if R is a commutative ring and P E R[x] is a polynomial whose
coefficients are zero divisors in R, then P is a zero divisor in R[x].
(iii) (Bonus) Prove or disprove: if R is a commutative ring and P is a zero divisor in R[x],
then every coefficient in P is a zero divisor in R.

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