Please help me with this question:
Let $n\in \Bbb N$, let $p$ be a prime number and let $\Bbb Z/ p^n\Bbb Z$ denote the ring of integers modulo $p^n$ under addition and multiplication modulo $p^n$. Let $f(x)$ and $g(x)$ be polynomials with coefficients from the ring $\Bbb Z/p^n\Bbb Z$ such that $f(x) · g(x) = 0$. Prove that $a_i b_j= 0$ for all $i,j$ where $a_i$ and $b_j$ are the coefficients of $f$ and $g$ respectively.
I thought that it was obvious at first, because I thought each of the coefficients of $f(x)g(x)$ were $a_ib_j$, but after some thought, I observed that coefficient may be of the form $a_1b_k+\ldots+a_kb_1$, and $f(x)g(x)=0$ only guarantees $a_1b_k+\ldots+a_kb_1$ is divisible by $p^n$. Then I thought that, maybe if $a_1b_k+\ldots+a_kb_1$ is divisible by $p^n$, then each term is divisible as well. But, $2+2$ is divisible by $2^2$ and yet 2 is not divisible by $2^2$.