Let $A = T_n(k)$ the set of upper triangular matrices be an algebra and let $k^n$ be a $A$-module given by left multiplication, I need find all submodules of $k^n$.
I know that if $N$ is an A-submodule of $k^n$ then $\frac{k^n}{N}$ has structure of $A$-module and the projection $\pi: k^n \to \frac{k^n}{N} $ should be $A$-linear.
My problem is that it is based on that idea, I don't know how to do to see specifically which are the submodules. Could you give me some help? Or if you can give me another idea of how to find the submodules I would also appreciate it.
Thanks
Let $e_1, \dots, e_n$ be the canonical basis of $k^n$. I assume that $T_n(k)$ is given by the set of all triangular matrices (not strictly triangular).
Let $W$ be a sub-$T_n(k)$-module of $k^n$. Let $I$ the smallest index such that $a_ie_i + \dots + a_ne_n \in W$ ($a_i \neq 0$). Then, acting by a diagonal matrix we get $e_i + w' \in W$. Using $E_{i,j}e_i = e_j$ if $j > i$ we get $w' \in W$ and hence $e_i \in W$. From this, it is easy to conclude that $W = \langle e_i, \dots, e_n \rangle $. Hence each $T_n(k)$-module is on this form.