$A$ is a $k$-algebra where $k$ is a commutative ring.
Then for $e$ an idempotent in $A$ and $U,V$ submodules of $Ae$ s.t. $Ae = U\oplus V$, there is unique idempotents $i, j$ in $A$ s.t. $U = Ai, V= Aj, e = i+j.$
The beginning of the proof goes:
Write $e = i + j $ for a unique $i \in U$ and $j \in V$ so we have $i = ij + i^2$. Then it is natural to conclude that $ij = 0, i^2 = i$ and $j^2 = j$.
Is there something off with the equality $i = ij + i^2$ as $e$ is not necessarily the unit of $A$ here? Any help would be appreciated!
Since $i \in U \subseteq Ae$, we can write $i=xe$ for some $x\in A$. Thus, $$ i = xe = xe \cdot e = i(i+j).$$