By definition a ring $R$ is Artinian if it is Artinian as $R$-module. I think the following is an example of an Artinian ring: $\mathbb Q / \mathbb Z$.
Ideals in it are of the form $(\frac1n)$ (since $(\frac{1}{n_1}, \dots , \frac{1}{n_k}) = (\frac{1}{\text{lcm}_i(n_i)})$).
Since we have $(\frac1n) \subset (\frac1m)$ if and only if $n$ divides $m$, every decreasing chain stabilizes eventually since $n$ only has finitely many divisors.
Is this correct?