Let $k$ be an algebraically closed field, let $R_p$ be the localization of $R=k[x,y]$ at the maximal ideal $p:=(x,y)$, i.e. $R_p:=\{F/G \in k(x,y) : g(0,0)\ne 0\}$. Then $R_p$ is a local ring with $m_p=\{f/g \in R_p : f(0,0)=0\}$ the unique maximal ideal. Now let $f,g\in k[x,y]$ be relatively prime, then $(f,g)_p $ is contained in $m_p$. My question:
Is it true that $\exists d>0$ such that $m_p^d \subseteq (f,g)_p$ ?
I think this is related with $k$ a field, $f,g$ be relatively prime polynomials in $k[x,y]$. Is every prime ideal of $k[x,y]$ containing $f$ and $g$ maximal?, but I can't figure out anything. Please help. Thanks in advance.