Show that there is a one-to-one correspondence between indempotents $e\in\operatorname{End}_R(V)$ and direct sum decompositions $V=X\dotplus Y$.
Attempt: We have $e^2 = e$. Therefore, we can write an element $x\in V$ as $x = e(x) + (1-e)(x)$. Write $X = Re$ and $Y = R(1-e)$. Then $V = X + Y$. Note that if $a$ belongs to $X \cap Y$, then $a = e(x) = (1-e)(y)$. So, $e(a) = e^2(x) = e(x)= a$ and $e(a) = e(1-e)(y) = -a$. Therefore $a= 0$. Hence $V$ decomposes into a direct sum.