Let $\Omega\subset\mathbb{C}$ be a disc. $p\in \Omega$.
In the integral domain $R=H(\Omega)$ " the space of holomorphic functions in $\Omega$, consider the maximal ideal $M_p=\{f \in R : f(p)=0\}$.
I need to find
$\cap_{n \geq 1}{M_p}^n$. (This intersection is the Jacobson ideal, isn't it?)
Here, ${M_p}^n$ is the nth power of $M_p$.
My thought, and according to the Krull Intersection Theorem, the intersection is the zero ideal. Even the theorem assumes that the ring is Notherian, but in my question $R$ is integral domain. Anyway, If $f \in \cap_{n \geq 1}{M_p}^n$ then $f \in {M_p}^n$ for all $n$. Does that mean that $f=0$ as $f(p)=\cdots =f^{(n)}(p)=0$? I think $\{{M_p}^n\}$ is increasing sequence in ideals.