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So I have been posed the problem of showing that $\begin{bmatrix} \mathbb{R} & \mathbb{R} \\ 0 & \mathbb{Q} \end{bmatrix}$ is left Artinian. Now, when showing rings are Noetherian, I usually show that every ideal is finitely generated. When showing rings are not Noetherian/Artinian I construct ascending/descending chains of ideals that do not stabilize. My issue really is that I have no idea how to go about proving that rings are Artinian (modulo obvious cases like fields).

So, although a hint about this particular matrix ring would be nice, I am really asking the more general question: What are the canonical techniques for proving that a ring is left/right Artinian?

  • Try to classify the ideals. In this case the image of any subspace under the map $\mathbb R^2\rightarrow\begin{bmatrix}\mathbb R&\mathbb R\0&\mathbb Q\end{bmatrix}$ is an ideal, and there are only 2 other ideals. – stewbasic Sep 29 '16 at 23:36
  • Related: https://math.stackexchange.com/questions/1532605 – Watson Dec 24 '16 at 13:07

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For this specific example

There's a method to find all one-sided ideals of a ring like this which I've described in this post and its links. It's a very useful thing to have under your belt.

Really it would probably be best to accomplish this problem using this method, or an ad-hoc hybrid with your own observations.

In general

Of course, there is no magic bullet for all problems. Aside from verifying the DCC on left ideals, there are a few other characterizations like these:

  1. $R$ has a generator $_RG$ which is Artinian.
  2. Every f.g. left $R$ module is Artinian
  3. Every f.cog. left $R$ module is Artinian

But they do not seem to be easier than verifying the chain condition directly.

There are more sophisticated combinations of theorems too, if you think you are justified in assuming them.

For example, Hopkins-Levitzki says that if you can prove the ring is left Noetherian, and you compute its radical and find it to be a nilpotent ideal, then it is also left Artinian.

For commutative rings, there is also another handy characterization: Artinian iff Noetherian and all primes ideals are maximal.

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