Identity of a ring as two different sums of idempotents

A simple example to show that this is not in general true without the orthogonality condition:

Let $R$ be a ring of characteristic $2$ with two conjugate but non-equal idempotents $e$ and $f$, so that $Re$ and $Rf$ are isomorphic as left $R$-modules. For example, take $R=M_2(\mathbb{F}_2)$, $e=\begin{pmatrix}1&0\\0&0\end{pmatrix}$ and $f=\begin{pmatrix}0&0\\0&1\end{pmatrix}$.

Then $$1=1+e+e+f+f=1+e+f+e+f.$$

As it is already commented by Mathematician 42 and san, this statement is usually stated with an orthogonality hypothesis (you can find, for example, Exercise 7 in here). So it is likely that your professor forget to mention about it.

You (and perhaps your professor) should be careful that the definition of an idempotent decomposition $e = e_1 + \dotsb + e_n$ requires not only $e_i^2 = e_i$ but also $e_ie_j = \delta_{ij}e_j$ (op. cit.), which I also had some trouble.

If you know that $e_ie_j=0$ for $i\ne j$ and $e′_ie′_j=0$ for $i\ne j$ (a condition forgotten to mention by the OP), then $a=\sum_i \phi_i^{-1}(e′_i)$ and $a^{−1}=\sum_i \phi_i(e_i)$ will do the job, where $\phi_i:R e_i\to Re′_i$ are the given left $R$-module isomorphisms.