Let me re-denote your chain complex $a$ by $C$. You can define a chain complex $D$ as the *mapping telescope* of the infinite sequence
$$
\cdots\overset e\to \quad C \quad \overset e\to \quad C \quad \overset e\to \cdots
$$
This can be constructed as follows:  Form the homotopy coequalizer of the pair of maps
$$
1,S_a: \bigoplus_{\Bbb N} \quad C \quad \to  \quad \bigoplus_{\Bbb N} \quad C 
$$
where $1$ is the identity and $S_a$ is given by applying $a$ and then shifting by one unit to the right in the index. (The homotopy coequalizer is gotten from this diagram by replacing the target 
$\oplus_{\Bbb N}  C$ with its cylinder $\oplus_{\Bbb N}  C \otimes I$ and forming the coequalizer of the two inclusions given by $1$ and $S_a$ on each end.)

The effect of this construction is to homotopically invert the map $e$, giving you a model for 
$C[e^{-1}]$.  There is an evident inclusion
$i: C \to D$. There is a map $D \to C$ which is defined on the $k$-th summand using the 
map $e^{\circ k}$. This will do what you want it to.