Are the following rings Artinian?

Question

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.

Answer 1

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