Splitting of an exact sequence 
Let $(R,\mathfrak m)$ be a Noetherian local ring. Suppose that $x \in \mathfrak m \setminus \mathfrak m^2$.  Is it true that
  $$
\frac{\mathfrak m}{x\mathfrak m} \cong \frac{\mathfrak m}{(x)} \oplus \frac{(x)}{x\mathfrak m}?$$

There exists an exact sequence
$$
0 \rightarrow \frac{(x)}{x\mathfrak m} \rightarrow \frac{\mathfrak m}{x\mathfrak m} \rightarrow \frac{\mathfrak m}{(x)} \rightarrow 0$$
and the question becomes: Does this sequence split? If not, what is a counterexample?
 A: In $\mathfrak m/\mathfrak m^2$ we have $\bar x\ne\bar 0$. Since $\mathfrak m/\mathfrak m^2$ is a finite dimensional $R/\mathfrak m$-vector space, there is a basis $\{\bar x,\bar x_1,\dots,\bar x_r\}$ in $\mathfrak m/\mathfrak m^2$. By NAK we have $\mathfrak m=(x,x_1,\dots,x_n)$.
Let $\mathfrak a=x\mathfrak m+(x_1,\dots,x_n)$.
Then $\mathfrak a+(x)=\mathfrak m$.
We also have $\mathfrak a\cap(x)=x\mathfrak m$. Let $y\in \mathfrak a\cap(x)$. Then $y=ax=a_0x+a_1x_1+\cdots+a_nx_n$ with $a_0\in \mathfrak m$, so $ax-(a_1x_1+\cdots+a_nx_n)=a_0x\in \mathfrak m^2$. It follows that $a\in \mathfrak m$ hence $y\in x\mathfrak m$. 
We get $\dfrac{\mathfrak m}{x\mathfrak m}=\dfrac{\mathfrak a}{x\mathfrak m}+\dfrac{(x)}{x\mathfrak m}$, and $\dfrac{\mathfrak a}{x\mathfrak m}\cap\dfrac{(x)}{x\mathfrak m}=(0)$. This shows that $\dfrac{(x)}{x\mathfrak m}$ is a direct summand in $\dfrac{\mathfrak m}{x\mathfrak m}$.
Furthermore, $\dfrac{\mathfrak a}{x\mathfrak m}\simeq\dfrac{(x_1,\dots,x_n)}{x\mathfrak m\cap(x_1,\dots,x_n)}$. But $x\mathfrak m\cap(x_1,\dots,x_n)=(x)\cap(x_1,\dots,x_n)$, so $\dfrac{\mathfrak a}{x\mathfrak m}\simeq\dfrac{(x_1,\dots,x_n)}{(x)\cap(x_1,\dots,x_n)}\simeq\dfrac{\mathfrak m}{(x)}$.
