Let $R$ be a ring, let $M$ be a left $R$-module, and let $e \in \operatorname{End}_R(M)$ satisfy $e \circ e = e$. Then
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\begin{align*}
M = \operatorname{im}(e) \oplus \ker(e).
\end{align*}
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Conversely, if $A,B \le M$ are submodules such that $M = A \oplus B$, and if $p: M \to M$ is the map defined by $p(a+b)=a$ for the unique decomposition $a \in A$ and $b \in B$, then $p \in \operatorname{End}_R(M)$, $p \circ p = p$, $\operatorname{im}(p)=A$, and $\ker(p)=B$.