Let be a ring with a proper left ideal such that, for any , either or is a unit.

(a) Prove that if is a left ideal which contains then either or .

(b) Prove that every proper left ideal of is contained in .

For (a) we have so we need to show that we can take an arbitrary and rearrange it so it looks like an element of . So if we take then for some and (because L is a left ideal). I'm not too sure how to proceed from here... how can we relate with ?

Thanks,