Generator of $F[x]$-module

Question

Suppose we have a field $F$ and a polynomial $g(x)$. Then $M= F[x]/(g(x))$ is a $F[x]$-module. I wonder what are the singletons that generate $M$. For example, $\{\bar 1\}$ generates $M$, since $F[x]\cdot\{\bar 1\}=M$. I think the singletons generate $M$ are exactly those $\{\bar f(x)\}$ such that $\gcd(f(x), g(x))=1$. Is this true?

Answer 1

If $f$ is a polynomial, then its class in $M$ generates the submodule $(f,g)/(g)$. As the ideal $(f,g)$ is generated by the gcd, you are right.