Let, $n\in \mathbb{N}$ and $p$ be a prime.
If $f(x)=\sum_{i=1}^{r} a_{i} x^{i}$ and $g(x)=\sum_{i=1}^{s} b_{i} x^{i}$ be polynomial in $\mathbb{Z}_{p^{n}}$ such that $f(x)g(x)=0$ ,
(1) $a_{i} b_{j} =0$,for all $i,j$
(2) $a_{i} b_{j} \neq 0$,for some $i,j$
My try, $a_{1} b_{1}=0 $
Either $a_1=0$ or $b_1=0$ or they are divisor of zero.
Then $a_{1} b_{2}+a_{2} b_{1}=0 $
Now, if $a_1=0$, then $a_{2} b_{1}=0 $
After, $a_{1} b_{3}+a_{2} b_{2}+a_{2} b_{3} = 0 $
If, $a_1=0$ then $a_{2} b_{2}+a_{2} b_{3} = 0 $
My thought here, if $\mathbb{Z_{p^n}}$ is $\mathbb{Z_2} $ here, then, I think option (2) is true if we take all of $a_2,b_2,b_3$ non zero.
But is option (1) the answer?
I am really in doubt here.
Please anyone help!
