Example of Artinian ring which is not a finitely generated k-algebra

In wikipedia it says:

Let $A$ be a commutative Noetherian ring with unity. Let $k$ be a field and $A$ finitely generated $k$-algebra. Then $A$ is Artinian if and only if $A$ is finitely generated as $k$-module.

Can anybody give me an example of Artinian ring which is a $k$-algebra but not finitely generated?

• I want an example that the Artinian ring is a $k$-algebra but infinitely generated – Bamqf Dec 17 '13 at 17:29

The real numbers over the rationals $\mathbb{Q}$ is a $\mathbb{Q}$-algebra, Artinian, but certainly not finitely generated.