The only maximal ideal of the set of all function germs around $p$

Question

Here is the definition we are using for the set of all function germs around p:

Now, I want to show that $m(p) := \{\bar{\phi} \in \mathcal{\varepsilon}(p)| \bar{\phi}(p) = 0\}$ is the only maximal ideal of $\mathcal{\varepsilon}(p).$

I know Proposition 12 in Dummit & Foote, third edition, which states that: Assume $R$ is commutative. The ideal $M$ is a maximal ideal iff the quotient ring $R/M$ is a field.

But still I do not know how to prove the required. Could anyone help me in this please?