A polynomial of degree $n$ over a commutative ring with zero divisors may have more than $n$ zeroes.
Attempt: Let $R$ be the commutatve ring which has a zero divisor $a \neq 0$. Then $\exists~~b \in R ,. b \neq 0$ such that $ab=0$.
We need to prove that $\exists~ f \in R[x]$ of degree $n$ such that $f$ has more than $n$ zeroes.
If we take the polynomial $ab x + ab^2 x^2 + ..... + ab^n x^n$ , this is essentially the zero polynomial as $ab=0$ and not a polynomial of degree $n$. So, I am not sure if this polynomial will work or not.
I am a bit confused. How do I move forward. Honestly, I am not sure if the problem statement is right itself.
Please note that my book has just introduced polynomial rings but none of reducability, irreducability, factorization etc.