I want to make the arrow in the numerator longer, like the one in the denominator and write $x$ on top of it. I'm using the package mathtools.
\[
H_{1}(K_1)=\frac{\textrm{ker}(R \xrightarrow{x} R)}{\textrm{im}(0 \longrightarrow R)} \cong \textrm{ker}(R \longrightarrow R)= \frac{R}{xR}.
\]


\[ ... \]is better than$$ ... $$for entering math mode. – ebosi Dec 12 '16 at 13:40\xrightarrow{~x~}or this\xrightarrow{\,\,x\,\,}– Steven B. Segletes Dec 12 '16 at 13:43