If $V$ is completely reducible and $End_{A}(V)$ is a division ring then $V$ is irreducible

Question

Since V is completely reducible we can write V as a direct sum of irreducible sub modules or a direct sum of 2 complement sub modules. Can anyone give a hint on which way to write V and how can I make use of $End_{A}(V)$ as a division ring.

I think writing $V = W \oplus U$ is better. Since W is any submodule. I think I should choose a map from V to W and show that W is equal to {0} or V

score 1 · Answer 1 · edited Dec 19 '20 at 01:30

1

Suppose that $V$ is not irreducible, i.e. $V=V_1\oplus V_2$.

Then $(Id_{V_1}\oplus0)\circ (0\oplus Id_{V_2})=0$. We deduce that $End_A(V)$ is not a division ring.

edited Dec 19 '20 at 01:30

anon

151,657

answered Dec 19 '20 at 01:14

Tsemo Aristide

87,475

0 is the zero function right? – Napu Dec 19 '20 at 11:52

If $V$ is completely reducible and $End_{A}(V)$ is a division ring then $V$ is irreducible

1 Answers1