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This question already has an answer here:

The following is an exercise in Hungerford (Ch. III, ex. 5.6).

Let $R$ be a commutative ring with identity. If $f=a_nx^n+\dots+a_0$ is a zero divisor in $R[x]$, then there exists a nonzero $b$ in $R$ such that $ba_n=ba_{n-1}=\dots=ba_0=0$.

I can see for example that $\{g\in R[x]\mid fg=0\}$ is a nonzero ideal, so it contains a nonzero element of smallest degree. But how to show that such an element is actually a constant?

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marked as duplicate by Daniel Fischer Apr 27 at 21:48

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

    
I gave here a complete solution to this question. – user26857 Dec 21 '12 at 21:58
up vote 4 down vote accepted

This is McCoy's theorem, for which you can find a nice summary and information on in this solution.

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