Need a little help with the following problem.This problem appears in Fulton's book on algebraic curves.The question is as follows:
Let $p=(0,0,...,0)\in \mathbb A^n$ and $\mathcal O_p=\mathcal O_p(\mathbb A^n)=\{f/g\in k(\mathbb A^n):g(p)\neq 0\}$ and $\mathbb m_p=\mathbb m_p(\mathbb A^n)=\{f\in \mathcal O_p:f(p)=0\}$ where $A()$ denotes the coordinate ring and $k()$ denotes the field of rational functions.Now suppose $I=\langle X_1,X_2,...,X_n\rangle\subset K[X_1,...,X_n]$,then $I\mathcal O_p=\mathbb m_p$.
My solution is as follows:
It is obvious that $\mathbb m_p\subset I\mathcal O_p$ because if $f\in \mathbb m_p$ then $f(p)=0$ and so $f\in I=\langle X_1,...,X_n\rangle$ and $1\in \mathcal O_p$ and hence $f\in I\mathcal O_p$.For the other way,let us take a general element from $I\mathcal O_p$ which looks like $f.\frac{\alpha}{\beta}$ where $\beta(p)\neq 0$ and $f(p)=0$ as $f\in I$ and thus $(f.\frac{\alpha}{\beta})(p)=0$ which gives us that the element is in $\mathbb m_p$.
I also think that it is quite obvious because $I$ is basically all the functions that vanish at $0$ and we are looking at the ideal generated by $I$ in the localization ring $\mathcal O_p$,so it should be nothing but $\mathbb m_p$ ,i.e. all the rational functions that vanish at $0$.
I am confused about the thing that $\mathbb m_p$ should be an ideal of $A(\mathbb A^n)$ but here the same $\mathbb m_p$ denotes an ideal in $\mathcal O_p$.
I am not sure whether I am making some mistake.Any help will be appreciated.