Commutative ring with no non-zero nilpotent elements

Modern Algebra
Ring Theory (XLIII)
Polynomial Rings over Commutative rings
Zero-divisor of a Commutative ring
Nilpotent Elements

Let R be a commutative ring with no non-zero nilpotent elements (that is, a^n = 0 implies a = 0).
If f(x) = a0 + a1x + a2x^2 +...+ amx^m in R[x] is a zero-divisor, prove that there is an element b ≠ 0 in R such that ba0 = ba1 = ba2 = ...=bam = 0.

See attached file for full problem description.

Attachments

Solution Summary

This solution is comprised of a detailed explanation of Zero divisor of Commutative Rings.
It contains step-by-step explanation of the problem that if f(x) = a0 + a1x + a2x^2 +...+ amx^m in R[x] is a zero-divisor,
then there is an element b ≠ 0 in R such that ba0 = ba1 = ba2 = ...=bam = 0 where R is a commutative ring with no non-zero nilpotent elements (that is, a^n = 0 implies a = 0). Notes are also given at the end.