I would like to solve the following:
By the Factor theorem a polynomial $f ∈ R[x]$, for $R$ is a field, then has a root in $R$ if and only if $(x-a)$ factor. Is the same statement necessarily true for $R[x]$ if $R$ is not a field? If not, provide a counter-example.
I know that the factor theorem is failed if $R$ is a non-commutative ring. So I was thing of the Quaternion. But I cannot figure out a counter-example.