Suppose I start out with the ring $\mathbb{Z}$, and call $\mathcal{C}$ the smallest collection of (commutative, unital) rings closed under the following list of operations (which I am aware has some redundancy):

- Localization
- Taking quotients
- Adding a variable: $A\mapsto A[X]$
- Completion with respect to an ideal
~~Tensor products~~~~More generally: finite limits and colimits~~

Since the tensor product of noetherian rings can be non-noetherian, I'm not even sure if everything in $\mathcal{C}$ is noetherian.

- Is there an easy-to-state characterization of which rings are in $\mathcal{C}$?
- Can much be said about what "nice" properties rings in $\mathcal{C}$ have? (I'm interpreting "nice" pretty loosely here).

I apologize in advance if this is well-known: even an answer of the form "this is all worked out in [X]" would be appreciated.

**Edit:** I would also be quite happy if anything could be said about $\mathcal{C}$ if we drop condition 6.

**Edit:** It looks like my naive hope that the objects of $\mathcal{C}$ might be "nice" is rather hopeless. If we drop conditions 5 and 6, then certainly everything in $\mathcal{C}$ is noetherian. Here is one way in which such rings might be called "nice":

If we define $\mathcal{C}$ using only 1-4, is the isomorphism problem for $\mathcal{C}$ decidable?