Zero divisors in polynomial rings

Question

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?

I gave here a complete solution to this question. – Dec 21 '12 at 21:58 — , Dec 21 '12 at 21:58

score 5 · Accepted Answer · edited Apr 13 '17 at 12:20

5

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

edited Apr 13 '17 at 12:20

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answered Dec 21 '12 at 21:31

rschwieb

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Zero divisors in polynomial rings

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