# Thread: Proof Help! Modules and left ideals

1. ## Proof Help! Modules and left ideals

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!

2. Originally Posted by dabien
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 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!

3. ## maximum rank in L

Hi, what do you mean by it is not accurate?
I was thinking about looking at the maximum rank in L,
but I still need help trying to figure this out.

Anything would be great!

4. ## help!

Ok I caught the error. But I am still very lost!

5. there's a quick way to solve your problem if you know a little bit about the "ring structure" of $\displaystyle R=M_n(F).$ what we need here is to know that $\displaystyle R$ is "semsimple" and thus every left ideal of $\displaystyle R$ is a

direct summand of $\displaystyle R.$ so if $\displaystyle L$ is a left ideal of $\displaystyle R,$ then $\displaystyle R=L \oplus K,$ for some left ideal $\displaystyle K$ of $\displaystyle R.$ therefore $\displaystyle 1=e+f,$ for some $\displaystyle e \in L, \ f \in K.$ thus for any $\displaystyle a \in L$ we have $\displaystyle a+0=a \cdot 1=ae + af,$

and $\displaystyle ae \in L, \ af \in K.$ thus $\displaystyle ae=a$ (and hence $\displaystyle af = 0,$ but we'll not use this one). this proves both $\displaystyle e^2=e$ and $\displaystyle L=Re.$