Let $f$ be a polynomial in $R[t]$ where $R$ is a ring. Let $\alpha \in R$. Is it true that $t - \alpha$ divides $f$ if and only if $f(\alpha) = 0$?
My thoughts on this matter (which isn't very well-formed yet because I'm still learning this stuff):
I see that this holds good if we substitute the ring $ R $ with field $ K $ then the above holds. I mean we can show that:
For $f \in K[t]$ where $K$ is a field and $\alpha \in K$, it is true that $t - \alpha$ divides $f$ if and only if $f(\alpha) = 0$?
The proof for this is long but at one place the proof depends on the fact that polynomials $ f, g \in K[t] $ ($ g \ne 0 $) can be written uniquely as $ f = qg + r $ such that $ q, r \in K[t] $ and $ deg(r) < deg(g) $.
The proof for this in turn depends on the fact that $ K[t] $ is an integral domain. But this proof does not work when $ R $ is an arbitrary ring (which may not be an integral domain). That leads me to the above question. Does this still hold good for $ R[t] $? If not, is there a counterexample?
