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!