Let $K$ be a field, $\partial\_i$ be commuting derivations on $K$, and consider the ring $R=K[\partial\_1\ldots \partial\_n]$ (it is implicitly assumed that the derivations do not commute with elements of $K$ and the commutation relation is the obvious $\partial\_i k - k\partial\_i=\partial\_i(k)$).


For example, $R$ can be the ring of differential operators with rational function coefficients, etc.


It appears that much of the theory of Groebner bases for polynomial rings $S=K[x\_1\ldots x\_n]$ can be trivially restated for $R=K[\partial\_1\ldots \partial\_n]$, since when we work with a monomial order, we usually track only what happens to the leading monomial, and it is the same for $R$ and $S$. I am pretty sure that this is also true for modules over $R$. (If I am wrong and this is not true, an explanation of this will be a perfect answer to this question:) )


Is there a reference which restates the standard results over $S$ for modules over $R$?


Most of the references I have found deal with the more(?) complicated case of polynomial coefficients. Other than that, I am aware of two references, both by Nobuki Takayama,
"Gröbner basis and the problem of contiguous relations" and a chapter in "Gröbner Bases: Statistics and Software Systems". The first basically introduces a version of Buchberger's algorithm for modules over $R$ and the second briefly goes over the very basic facts for ideals in $R$.



Groebner bases for differential operators with field coefficients (reference request)
Elaborating on Michael's comments and your concerns about Saito, Sturmfels, Takayama, "Grobner Deformations of Hypergeometric Differential Equations", the case this book addresses is the case of the Weyl algebra over a characteristic 0 field $k$. I.e., the noncommutative polynomial ring $$D=k[x\_1,\ldots, x\_n, \partial\_1,\ldots, \partial\_n]$$, subject to the commutator relations $[\partial\_i, x\_j]=\delta\_{ij}$. There's also a few points at which they address the larger ring $$k(x\_1,\ldots, x\_n)\otimes\_{k[x\_1,\ldots, x\_n]} D.$$ Is this what you're looking for?


As for online things, some of the above is also covered in Oaku and Takayama's "Algorithms for D-modules —restriction, tensor product, localization, and local cohomology groups".


