I'm trying to prove the following statement.
Suppose $ L \lhd R$ is a left ideal in $ R $ (a ring with unity). If there exists a left ideal $ N \lhd R $ such that $ R = L \oplus N $, then $ L = Re $ for $ e \in R $ satisfying $ e^2 = e $.
I don't really know where to begin with. Those two statements don't seem connected in any way to me. I would appreciate any hints
There exist unique pair $l\in L, n\in N$ such that $l+n=1$. But, by the definition of ideal, $ln$ lie in both ideals, so it must be zero, since we have a direct sum. So, we get $l(1-l)=0$, i.e. $l^2=l$. Now, since for any element $a$ of the ring, we have $a=1a=la+(1-l)a$, and also we know that any element can be uniquely written as such a combination, we conclude that $L=Rl$ for this $l$.