Here is the question:
Show that if L is a left ideal of the ring M2(F), F any field, then L consists of all left multiples of some element A in M2(F).
Show that this A can be chosen to be idempotent, such that A^2=A.
Show that the foregoing assertions hold not just for M2(F) but also for Mn(F), any n.
Any help would be great!
there's a quick way to solve your problem if you know a little bit about the "ring structure" of what we need here is to know that is "semsimple" and thus every left ideal of is a
direct summand of so if is a left ideal of then for some left ideal of therefore for some thus for any we have
and thus (and hence but we'll not use this one). this proves both and