Let $\mathcal{E}_n$ be the algebra of germs at the origin of smooth functions $f : \mathbb{R}^n \supset U \to \mathbb{R}$, where $U$ is an open set containing $0$, and let $M_n \subset \mathcal{E}_n$ be the (maximal) ideal of germs that vanish at $0$.
An exercise in Brocker and Janich's Introduction to Differential Topology begins: "$\mathcal{E}_n/M_n \cong \mathbb{R}$; consequently $M_n/M_n^2 \cong \mathbb{R}^n$. Show that..."
Given that these two claims are made before the "problem" part of the problem, I figure they must be pretty immediate. I was able to convince myself that $\mathcal{E}_n/M_n \cong \mathbb{R}$, but I can't figure out why the second identification is true, or how it follows from the first. I don't need a full solution; all I'd like is to know if it follows from a basic property of algebras/rings, or if it requires the specific properties of the algebras and ideals under consideration.
The third isomorphism theorem says:
If $I,J$ are ideals of a ring $R$ such that $I\subseteq J\subseteq R$, then $R/J\cong (R/I)/(J/I)$