not an Ideal of F[x] Let $F$ be a field, and $R$ be the polynomial ring $F[x]$, and $f,g,d\in R$ be monic. Let $\Gamma=$ {monic $d\in R : d|f$ and $d|g$}. Show $\Gamma$ is not ideal, and it has a unique element of maximal degree: $gcd(f,g)$. I don't even know where to begin :(.
First let $d \in R$ be an element of $R$ which is a divisor of $f$ and $g$. Now find an element $r$ in $R$ for which $d*r$ or $r*d$ is not in $\Gamma$.
First let $d \in R$ be an element of $R$ which is a divisor of $f$ and $g$. Now find an element $r$ in $R$ for which $d*r$ or $r*d$ is not in $\Gamma$.