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# Commutative ring with no non-zero nilpotent elements

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Let R be a commutative ring with no non-zero nilpotent elements (that is, a^n = 0 implies a = 0).
If f(x) = a_0 + a_1x + a_2x^2 +...+ a_mx^m in R[x] is a zero-divisor, prove that there is an element b is not equal to 0 in R such that ba_0 = ba_1 = ba_2 = ...=ba_m = 0.

See attached file for full problem description.

https://brainmass.com/math/ring-theory/commutative-ring-no-non-zero-nilpotent-elements-128122

#### Solution Preview

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

By:- Thokchom Sarojkumar Sinha

Let be a commutative ring with no non-zero nilpotent elements ( that is, implies ).
If in is a zero-divisor,
prove that there is an element in such that .

Solution:- Let be a zero-divisor .
Then there exists non-zero polynomial such that

where

Then ------------------------------------------------------------------------(1)

If then

For is a commutative ring.
by using (1).
-----------------------------------------------------------------(2)
and
For if since has no non-zero nilpotent elements.

Again,

For is commutative.
by using (1).

For by (2).

...

#### 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) = a_0 + a_1x + a_2x^2 +...+ a_mx^m in R[x] is a zero-divisor,
then there is an element b is not equal to 0 in R such that ba_0 = ba_1 = ba_2 = ...=ba_m = 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.

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