Let $R$ be the ring defined by $$R=\left\{\begin{pmatrix}a&0\\c&d\end{pmatrix}\mid a\in\mathbb{Q},c,d\in\mathbb{R}\right\}.$$ I want to show that it is left-artinian. I have calculated all possible left ideals: $$L_1=\begin{pmatrix}0&0\\0&0\end{pmatrix} $$ $$L_2=\begin{pmatrix}\mathbb{Q}&0\\\mathbb{R}&\mathbb{R}\end{pmatrix}$$ $$L_3=\begin{pmatrix}0&0\\\mathbb{R}&0\end{pmatrix} $$ $$L_4=\begin{pmatrix}0&0\\0&\mathbb{R}\end{pmatrix} $$ $$L_5=\begin{pmatrix}\mathbb{Q}&0\\\mathbb{R}&0\end{pmatrix} $$ $$L_6=\begin{pmatrix}0&0\\\mathbb{R}&\mathbb{R}\end{pmatrix} $$ $$L_7=\mathbb{R}\begin{pmatrix}0&0\\1&r\end{pmatrix} $$for $r \in\mathbb{R}.$ How can I conclude that $R$ is left artinian?
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1See this post. See also this post, for the continuation of this exercise. – Dietrich Burde Nov 04 '22 at 19:44
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@DietrichBurde could I use the following: $I_v:=\mathbb{Q}\pmatrix{0&0\v&0}$ for $v\in\mathbb{R}$. Then $I_0, I_{\pi}, I_{\pi^2}, ...$ is doing the job? – hannah2002 Nov 04 '22 at 19:48
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1In the future, you ought to accumulate your work on a problem in a single post, not delete and repost the problem as you get feedback. This will be construed as dodging moderation (and kind of wasting everyone's time, too.) – rschwieb Nov 04 '22 at 20:05