$(R, k, \mathfrak{m})$ is a noetherian local ring.
Consider a short exact sequence of $R-$modules
$0 \to k \xrightarrow{\alpha} B \to C \to 0$
in which $\alpha(1) = e,$ where $e \in B - \mathfrak{m}B$. In particular, when $B = \mathfrak{m}/x\mathfrak{m}$ and $k $ is the cyclic submodule generated by $e + x\mathfrak{m}$. Here $x \in \mathfrak{m} - x\mathfrak{m}^2$ is a nonzero divisor.
I am not sure how $k $ is the cyclic submodule generated by $e + x\mathfrak{m}$. Any help would be appreciated!
We have $k=R/\mathfrak{m}$, so for all $m\in \mathfrak{m}$ we have $1{m}=0$. Thus for all $m\in \mathfrak{m}$ we have: $$em=\alpha(1)m=\alpha(1m)=\alpha(0)=0.$$
Thus $\langle e\rangle$ is well defined as a vector space over $R/\mathfrak{m}=k$. However it is generated by a single element so must be $k$ or $\{0\}$. It cannot be $\{0\}$ by exactness of the s.e.s. so we have $\langle e\rangle=k$.