Let $k$ be a field and let $A$ be a (commutative) $k$-algebra. Assume that for every maximal ideal $P \subseteq A$ the residue class field $A/P$ has finite dimension as a $k$-vector space.
Is there a name for $k$-algebras like that?
Clearly, finitely generated $k$-algebras $A$ satisfy this property, but what about the case $A$ not finitely generated over $k$?
Examples of such algebras can easily be obtained by localizing finitely generated $k$-algebras:
For instance let $\mathcal{O} = k[x]_{(x)}$ be the localization of the finite $k$-algebra $k[x]$ by the maximal ideal generated by $x$. Then $\mathcal{O}$ is a local ring with maximal ideal $P$ generated by $x$ which is a non finitely generated $k$-algebra. But it has a finite dimensional residue class field $\mathcal{O}/P \cong k$.
I was thinking about this since I am working with schemes over $k$ that do not need to be of finite type over $k$ but have finite dimensional residue class fields as above. I haven't encountered a definition or naming of this.
I am grateful for any kind of input. I posted this question also on math.stackexchange, but did not receive any answer or input.
