Artinian rings that are not Artin algebras An Artin algebra $A$ is an algebra over a commutative Artinian ring $R$ which is finitely generated over $R$, e.g. finite dimensional algebra over a field. Clearly, any Artin algebra is left and right Artinian. 
What are examples of left and right Artinian rings that not Artin algebras?
 A: Consider the first Weyl field $D_1(k)$ over a field $k$, that is, the skew-field of quotients of the first Weyl algebra $A_1(k)$ over $k$. As it is a division ring, it is of course left and right artinian. On the other hand, the center of $D_1(k)$ is $k$, so $D(k)$ is not finitely generated  as a module over any central subring — if $k=\mathbb  C$ for example, $D_1$ has uncountable dimension over $k$.
The first example in history of such a thing is as follows: let $\def\QQ{\mathbb{Q}}\QQ(t)$ be the field of rational functions with rational coefficients, let $\sigma:\QQ(t)\to\QQ(t)$ be the field automorphism such that $\sigma(t)=2t$, consider the ring $\QQ(t)[[X]]$ of power series with coefficients in $\QQ(t)$ with multiplication twisted so that $Xf=\sigma(f)X$ for all $f\in\QQ(t)$ (this can be viewed as a power-series version of the Ore extension of $\QQ(t)$ by $\sigma$) One can localize this ring by inverting $X$, and one gets a ring of Laurent series $D$ with coefficients in $\QQ(t)$ in the variable $X$ with the same commutation relation. The center of $D$ is $\QQ$, so $D$ is not finitely generated over its center. This example is due to Hilbert, if I recall correctly. 
