# 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.

#### 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.