For the upper triangular matrix ring given by $\begin{pmatrix}\mathbb{Q}&\mathbb{R}\\0&\mathbb{R}\end{pmatrix}$ I want to determine its left ideals. For example, this link refers to Lam's "A first course in non-commutative rings" applied to a generic matrix ring $\begin{pmatrix}R&M\\0&S\end{pmatrix}$:
The left ideals are all of the form $I_1\oplus I_2$ where $I_2$ is a left ideal of $S$, and $I_1$ is a left $R$ submodule of $R\oplus M$ which contains $MI_2$.
I want to translate this to explicitly determine all the left-ideals of my matrix ring above. So this is just a mere translation exercise. Let's see if I can do it correctly.
$I_2$ is a left ideal of $\mathbb{R} \implies I_2 = 0$ or $I_2 = \mathbb{R}$
$I_1$ is a left $\mathbb{Q}$-submodule of $\mathbb{Q} \oplus \mathbb{R}$ which contains $\mathbb{R}I_2$
So now, I unroll this complex definition.
A $\mathbb{Q}$-module is a vector space over $\mathbb{Q}$. The underlying set for this vector space is $\mathbb{Q} \oplus \mathbb{R}$ which I identify with $\mathbb{Q} \times \mathbb{R}$. So here I'm considering operations of the kind $q_1(q,r) = (q_1q,q_1r)$ and I have to determine the subspaces of this vector space. Is it as easy as taking a subspace of $\mathbb{Q}$ (probably $\{0\}$ or $\mathbb{Q}$ and a subspace of $\mathbb{R}$ over $\mathbb{Q}$ (giving a field extension?) and then doing the product? But then, how can this contain $\{0\},\mathbb{R}$ if it is bidimensional?
Can you help me to figure out this situation?
Thoughts
There is some sort of identification in the statement of the theorem because $MI_2$ should be $\mathbb{R}$ not $\{0\} \times \mathbb{R}$
ok, now I see the identification, it corresponds to $\begin{bmatrix}0 & M \\ 0 & 0\end{bmatrix}\begin{bmatrix}0 & 0 \\ 0 & I_2\end{bmatrix} = \begin{bmatrix}0 & MI_2 \\ 0 & 0\end{bmatrix}$
However I still don't see very well what should be explicitely the sets in @rschwieb answer. It seems to me that for the second one the possibilities are just $\mathbb{Q} \times \mathbb{R}$ and $\{0\} \times \mathbb{R}$. What about the first one?