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I am trying to figure out whether the following ring is artinian:

$R = \left\{\left( \begin{array}{cc} a & b \\ 0 & a \\ \end{array} \right)\mid a,b \in \mathbb{C}\right\}$.

I understand that if $\mathbb{C}$ were instead replaced by a finite field then $R$ would be artinian as it is finite. However, I am struggling to see whether this is still the case when finite is replaced by infinite.

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    The accepted answer to [this question](http://math.stackexchange.com/q/1567500/110831) is also an answer to yours. – 2017-02-18

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Hint: a left (right) ideal is a vector subspace.