I wonder if a category of constructible rings does make sense and how it differs from the complete category of rings resp. how it could be completed.
Let the objects $\mathcal{O}$ of this category be recursively defined by
- $\mathbb{C} \in \mathcal{O}$
- $\mathcal{R} \in \mathcal{O}$ and $\mathcal{R}'$ a subring of $\mathcal{R}$ $\rightarrow $ $\mathcal{R}' \in \mathcal{O}$
- $\mathcal{R} \in \mathcal{O}$ and $n \in \mathbb{N}$ $\rightarrow $ $M_n(\mathcal{R}) \in \mathcal{O}$ (matrix ring)
- $\mathcal{R} \in \mathcal{O}$ and $n \in \mathbb{N}$$\rightarrow $ $\mathcal{R}[X_1,\dots,X_n] \in \mathcal{O}$ (polynomial ring)
- $\mathcal{R} \in \mathcal{O}$ and $\mathcal{R}$ is an integral domain $\rightarrow $ $\operatorname{Quot}(\mathcal{R}) \in \mathcal{O}$ (field of fractions)
- $\mathcal{R} \in \mathcal{O}$ and $I$ is a two-sided ideal in $\mathcal{R}$ $\rightarrow $ $\mathcal{R}/I \in \mathcal{O}$ (quotient ring)
- $\mathcal{R} \in \mathcal{O}$, $\mathcal{R}$ a subring of $\mathcal{R}'$ and $x \in \mathcal{R}'$ $\rightarrow $ $\mathcal{R}[x] \in \mathcal{O}$ (extension)
Is this list of constructions complete in a way? Which other constructions are relevant for building up a category of constructible rings? The other way around:
Which "natural" rings would one expect to be constructible but could not be constructed by the rules above?
I assume that by these constructions the p-adic rings $\mathbb{Z}_p$, $\mathbb{Q}_p$ would not be in the category of constructible rings, while $\mathbb{R}$ is because it's a subring of $\mathbb{C}$.
Can in a way a completed category of rings be obtained from a restricted category of constructible rings by adding all "limits"?
(I've learned that $\mathbb{Z}_p = \lim_{\leftarrow} \mathbb{Z}/p^k\mathbb{Z}$ where all quotient rings $\mathbb{Z}/p^k\mathbb{Z}$ are in $\mathcal{O}$.)
$\mathbb{Q}_p$ is in your category for a fairly non-obvious reason: actually it already arises in the very second step, as a subring of $\mathbb{C}$. $\mathbb{C}$ has a lot of subrings. In fact:
Proof. If $D$ is such a domain, the algebraic closure of its fraction field is an algebraically closed field of characteristic zero which is at most uncountable, and up to isomorphism, algebraically closed fields of characteristic zero are determined by their cardinality. In particular there is a unique countable one, namely $\overline{\mathbb{Q}}$, and a unique one with cardinality $\mathbb{C}$, namely $\mathbb{C}$ itself. $\Box$
On the other hand, none of your constructions increase the cardinality of a ring beyond the cardinality of $\mathbb{C}$, so your category does not include any ring with a larger cardinality. I know of no really natural examples of such rings but we can take, for example, the ring $\mathbb{R}^{\mathbb{R}}$.
A speculation: among the steps you've listed, only the construction of matrix rings makes a ring "more noncommutative," and for that reason I think, although I'm not sure, that "very noncommutative" rings such as the Weyl algebra can't be constructed this way.
The most important construction which doesn't explicitly appear on your list, although it can be done to some extent using the constructions on your list, is probably localization.
As for the more general question of how to generate rings, for general categorical reasons, every ring can be constructed from the ring $\mathbb{Z}[x]$ by repeatedly taking colimits. In fact it suffices to be able to take coproducts, which here are free products, and coequalizers. Somewhat more explicitly, every ring is a quotient of a noncommutative polynomial ring on a sufficiently large number of generators by a suitable two-sided ideal, which is to say that every ring has a presentation by generators and relations, just as for groups. There are other ways of splitting this up; for example, again for general categorical reasons, every ring is a filtered colimit of finitely presented rings.