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Let $R[x]$ be a polynomial ring. Show that if $R$ is finite and has zero divisors, $R[x]$ has an infinite number of zero divisors.

I'm having trouble wrapping my head around what exactly polynomial rings are. And hints pointing me in the right direction would be appreciated.

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Do you have an abstract algebra textbook? If so, the definition of a polynomial ring can surely be found within it. If you have already done so, then you should be more specific as to what about polynomial rings it is that you do not understand. – John Martin Nov 17 '12 at 23:14
Its not the definition I'm confused about, its the concept of a what a zero divisor would look like in a polynomial ring. – Carly Nov 17 '12 at 23:15
@Carly Any zero divisors in $R$ could be zero divisors in $R[X]$, understood as constant polynomials, among others. – Cocopuffs Nov 17 '12 at 23:20
up vote 1 down vote accepted

If $r$ is an element of $R$ such that $r$ is a zero divisor, then $r$ is an element of $R[x]$, and $rx, rx^2, ..., rx^n,...$ are all elements of $R[x]$, and since $\Bbb N$ is infinite, there is an infinite number of zero divisors.

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