# Ideal of $\text{End}_k V$ has certain form.

Let $V$ be a finite-dimensional vector space over field $k$ and $R = \text{End}_k V$. How do I see that any left ideal of $R$ takes on the form $Rr$ for some suitable element $r \in R$?

#### Solutions Collecting From Web of "Ideal of $\text{End}_k V$ has certain form."

Since $R$ is a semisimple ring, every left ideal will split out like this: $R=L\oplus L’$. In this decomposition, $1=e+e’$ where $e\in L$ and $e’\in L’$.

It’s an exercise to prove that $e^2=e$ and $(e’)^2=e’$. (Really, $e’=1-e$.)

It’s another exercise to show that $L=Re$. Just remember that the sum is direct.