Endomorphism rings of a $k$-algebra Let $k$ be a field and $R$ be the $3$-dimensional $k$-algebra $\left [\begin{array}\ k & k \\ 0 & k \end{array} \right ]$. I have two conjectures:
(1) Is any simple right $R$-module has endomorphism ring $k$? (Namely, is any endomorphism $f$ a scalar multiplication by a fix element of $k$?)
(2) Is any proper right ideal of $R$ of length at most two?
Thanks in advance!

It has two isoclasses of simple right modules. As you mentioned in the comments, there are those two maximal ideals. Those are the only maximal ideals since the maximal ideals correspond to the maximal ideals of $k\times k$, of which there are just two.  So you have answered that part of the question yourself.
The section assertion is trivially true because the composition length of the ring as a right module is $3$, and composition length is additive, that is:
$\ell(M)=\ell(N)+\ell(M/N)$ (whenever composition length is defined for $M$ and $N$.)
