TITLE: Example of Artinian ring which is not a finitely generated k-algebra
QUESTION [5 upvotes]: 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?

REPLY [6 votes]: The real numbers over the rationals $\mathbb{Q}$ is a $\mathbb{Q}$-algebra, Artinian, but certainly not finitely generated.