Is there any notion of algebraic closure for commutative rings? I am specifically interested in such a concept for $\mathbb Z_n$, with $n$ not a prime (possibly square-free).
Such a concept would be easy to develop for integral domains since these can be naturally embedded in their own fraction field. But what about rings having divisors of $0$?
I am also aware of the concept of integral closure of a ring $A$ in $B$, but this does not suit me because, unlike the algebraic closure for fields, the integral closure requires me to choose some auxiliary $B$, so this is a "relative" concept, not an "absolute" one.
If $n$ is square-free, could the fact that my ring is a product of fields simplify the job?
One way to describe the algebraic closure is that it is in some sense a "maximal" algebraic extension: it's an algebraic extension into which every other extension embeds. So it seems to me like the following question is a more basic one that should be answered first:
There are various ways to answer this question depending on what you're trying to do. The simplest answer, generalizing finite extensions, is finite morphisms of rings; this means that the map $f : A \to B$ exhibits $B$ as a finitely generated $A$-module. There are various notions of Galois extension for commutative rings, and various corresponding Galois theories. And there is the notion of an étale morphism, which leads to a Galois theory involving the étale fundamental group.
I think with most of these definitions you'll run into the problem that there will in general not be a "maximal" such extension.