TITLE: Submodules of an $R$-module where $R$ is the set of $n\times n$ upper triangular matrices
QUESTION [5 upvotes]: Let $R$ be the ring of $n\times n$ upper triangular matrices with coefficients in a field $K$. Let $V$ be the $R$-module consisting of all $1\times n$ matrices with coefficients in $K$. Define $$V_r = \{(x_1,\dots,x_n) \in V\mid x_i = 0 \text{ for all } i \text{ less than or equal to } r\}.$$ How can I show that $V_0, V_1,\dots,V_n$ are the only $R$-submodules of $V$?

REPLY [3 votes]: It looks like you are operating on the right of $V$ by matrix multiplication. I'll use $e_i$ to denote the thing in $V$ which is all zeros except for an 1 in the $i$th position.
Hint-lemmas:
Lemma 1: Prove that if $x\in V$ is nonzero and one of its coefficients $x_i$ is nonzero, then there is a $r\in R$ such that $xr=e_i$.
Lemma 2: Prove that if $N$ is a submodule of $V$ and $e_i\in N$, then $e_j\in N$ for $j\geq i$.
Lemma 3: Note that the modules $V_r$ are linearly ordered.
Combine these lemmas to argue that an arbitrary submodule is contained in, and then equal to, one of the $V_r$.