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In all the texts I have seen on differential algebra, differential rings/fields/algebras/etc. are always specified as having a finite number of derivations (some books e.g. those specifically on differential Galois theory, even introduce the differential rings/etc. as having only a single derivation). I am somewhat confused why the finite restriction is hammered in place.

There are a few results in Kolchin's book that are proved using induction on the number of derivations. There is also at least one proof I could find while skimming back that uses a matrix over a vector space whose basis is the set of derivations. But as for the induction proofs, they could probably still work through some careful application of abstract nonsense or transfinite induction. I'm not terribly familiar with the logistics of infinite matrices, but perhaps the matrix proof(s) would still work by considering those.

More than anything though, the results that are proved in this fashion seem to be quite few in between. It seems like the vast majority of results could be extended to the infinite case, and then one could simply add in the other theorems as special cases that only apply to "finite-partial" differential rings.

Is it simply a problem of few applications of rings/etc. with an infinite set of derivations? Or is there some very important result that actually doesn't hold in "infinite-partial" differential rings, that makes studying them unfruitful? Or is there actually a body of research in this direction that I simply am not aware of?

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I do not know which results proofs you refer to. So I cannot say whether these results hold for an infinite number of variables. But let me mention the following the following generalization.

You can describe a differential algebra $R$ in one variable over a commutative base ring $k$ also in the following form:

Let $k[x]$ be the polynomial algebra in one variable $x$ over $k$ and make $k[x]$ into a $k$-bialgebra with comultiplication $\Delta$ defined by $\Delta(x) = x \otimes 1 + 1 \otimes x$ and counit defined by $\varepsilon(x) = 0$.

A derivation on a commutative $k$-algebra $R$ can also be described as a morphism of $k$-modules $\psi : k[x] \otimes_k R \to R$ such that $\psi(d \otimes ab) = \sum_{(d)} d_{(1)}(a) d_{(2)}(b)$ for all $d \in k[x]$ and $a,b\in R$ and such that $\psi(d \otimes 1) = \varepsilon(d) 1$. The coalgebra structure mentioned above models the properties of a derivation. Such a morphism $\psi$ is called a measuring and you can define a measuring for any coalgebra $D$ instead of $k[x]$. If $D$ is moreover a bialgebra then a $D$-module algebra is a $k$-algebra $R$ with a measuring $\psi$ as above such that $\psi$ makes $R$ into a (left) $D$-module. (Speaking more abstractly, this is a monoid in the monoidal category of $D$-modules. This explains the name.)

Several results from differential algebra generalize to this setting. For instance the Picard-Vessiot theory has a natural generalization in this setting. See:

Katsutoshi Amano and Akira Masuoka. Picard-Vessiot extensions of Artinian simple module algebras. J. Algebra, 285(2):743–767, 2005.

In order to deal with differential rings with several derivations, you can consider polynomial algebras in several variables, one for each derivation, with coalgebra structure described as above (for each variable).

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