Left Ideals in $C^*$ algebra

Question

Let $A$ be $C^*$-algebra, $L$ be a maximal left ideal, $\Lambda(A)= \{ L \subset A\}$, and $rad(A) = \cap_{L \in \Lambda(A)} L$.

I'm trying to understand this Lemma. But I'm stuck on understanding these points.

1. Why no left ideal admits $e$?

2. Why we can use Zorn's lemma here?

Thank you in advance!