Let $\mathbb{k}$ be an algebraically closed field and let $A$ be a $\mathbb{k}$-algebra which is a free module of rank $r$ over some central subalgebra $Z_0$. If $Z_0$ is affine and $r$ is a finite number then the maximal ideals of $Z_0$ all have $\mathbb{k}$-codimension 1 and so the quotients $$\{A / \mathbb{m} A \mid \mathbb{m} \in \operatorname{Max}(Z_0)\}$$ all have dimension $r$ over $\mathbb{k}$. One could say that an algebraic property is generic for this family of algebras if there is a non-empty Zariski open subset $\mathcal{U}$ of the algebraic variety $\operatorname{Max}(Z_0)$ such that for all $\mathfrak{m} \in \mathcal{U}$ the algebra $A/ \mathfrak{m} A$ possesses this property. There are many results which state that certain properties are generic for certain families of algebras. For example, it is not hard to prove that the centre of $A/\mathfrak{m} A$ attains its minimal value on a Zariski open subset of $\operatorname{Max}(Z_0)$. I believe that if $A/\mathfrak{m}A$ is a simple algebra for some $\mathfrak{m}$ then the same is true generically, although I don't know a proof, perhaps other hypotheses are needed. As a side question, does anyone know a proof for this claim?
I want to understand some examples where $A$ has countable rank as a free $Z_0$-module, and $Z_0$ is a countably generated central subalgebra of $A$. Provided the cardinality of the field is uncountable or greater the maximal ideals of $Z_0$ have codimension 1 over $\mathbb{k}$ and so we have a reasonable description of $\operatorname{Max}(Z_0)$. What can be said about generic properties of such families? Is there an obvious replacement for the Zariski topology on $\operatorname{Max}(Z_0)$? This is the first obvious question if you want to claim that some property is generic in this case. Are there any known results about generic properties for such families of algebras?
I'd be especially interested in ideas which work over fields of characteristic $p > 0$ but obviously I'm interested to hear comments which only apply to $\mathbb{C}$, for example.
My interest in this question came from considering representations of countable dimensional restricted Lie algebras: the enveloping algebra of such a Lie algebra $\mathfrak{g}$ in characteristic $p > 0$ is a free module over the $p$-centre, and the latter is a countably generated polynomial ring.
