Determine the left ideals of the matrix ring $A \subseteq M_2(\mathbb{R})$ where $$A=\{\begin{pmatrix} \mathbb{R} & \mathbb{R}\\ 0 & \mathbb{R} \end{pmatrix}\}$$ and the multiplication is the usual matrix multiplication. It looks to me as if there are only 4 ideals. Basically the matrices spanned by \begin{pmatrix} \mathbb{0} & \mathbb{R}\\ 0 & \mathbb{R} \end{pmatrix}, \begin{pmatrix} \mathbb{R} & \mathbb{0}\\ 0 & \mathbb{0} \end{pmatrix} and \begin{pmatrix} \mathbb{R} & \mathbb{R}\\ 0 & \mathbb{0} \end{pmatrix} respectively and perhaps the ideals $\{0\}$ and $A$. I just want to be sure I'm not missing any. All feedback is appreciated.
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See this post. – Dietrich Burde Feb 28 '22 at 17:57
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For a fixed $\lambda\neq 0$, the set of matrices of the form $\begin{bmatrix}x&\lambda x\0&0\end{bmatrix}$ is a left ideal, and there are infinitely many of that form. If you take a look at the duplicate, this corresponds to the case of $I_2={0}$ and $I_1$ just being some subspace of $\mathbb R\times\mathbb R$. – rschwieb Feb 28 '22 at 18:22