I have a question, more about how to calculate the kernel of a certain map:
Let the ring $\varepsilon_n=\lbrace f\colon (\mathbb{R}^{n},0)\longrightarrow \mathbb{R}: f \quad\textit{is map germ}\rbrace$ and maximal ideal $\mathcal{M}_{n}=\lbrace f\in\varepsilon_{n}:[f]_{0}=0\rbrace$, and I have also shown that
\begin{align*} \mathcal{M}_{n}&=\langle x_1,\ldots x_n\rangle\\ \mathcal{M}_{n}^{k}&=\langle x_{1}^{i_1},\ldots,x_{n}^{i_n}:i_1+\cdots +i_{n}=k\rangle \end{align*}
in the same way let's define $\overset{\sim}{\varepsilon_{n}}$ as the set of taylor's series
\begin{equation*}
\overset{\sim}{f}=f(0)=f(0)+\frac{1}{1!}\sum\frac{\partial f}{\partial x_i}(0)x_{i}+\frac{1}{2!}\sum\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}(0)x_{i}x_{j}+\cdots
\end{equation*}
Let
\begin{align*}
J\colon\varepsilon_n &\longrightarrow \overset{\sim}{\varepsilon_n}\\
\hspace{4.1mm} f&\longmapsto \overset{\sim}{f}
\end{align*}
where $(2.4)$ is
which is what I indicated at the beginning, this is an excerpt from the book Singularity points of smooth mapping-Gibson
I don't know if it is very trivial, but I am a bit confused, I must show that the $$ker(J)=\bigcap_{k=1}^{\infty}\mathcal{M}_{n}^{k}$$ any questions or solutions are welcome.