TITLE: How much theory works out for "almost commutative" rings?
QUESTION [16 upvotes]: I've been reading about D-modules, and have seen a proof that D_X, the ring of differential operators on a variety, is "almost commutative", that is, that its associated graded ring is commutative.  Now, I know that with the Ore conditions, we can localize almost commutative rings, and so we get a legitimate sheaf D to do geometry with.  But how far does the analogy go? What theorems are true for commutative rings but can't be modified reasonably to be true for (nice, say, left and right noetherian or somesuch) almost commutative rings?

REPLY [3 votes]: Check out the paper differential operator on noncommutative ring, Rosenberg-Lunts define the noncommutative version of Grothendieck differential operator. This framework works perfect for arbitrary noncommutative ring. As an application. They developed quantum D-module theory as a special case of Noncommutative D-module theory. They treated the much more general case in Differential Calculus in noncommutative algebraic geometry
As Greg mentioned above, considering example of Weyl algebra. In commutative geometry, there is not good notion for closed subscheme(at least for diagonal). So,the crucial step is to find what is the correct notion for closed subscheme for noncommutative ring which led to the definition of noncommutative subscheme(naturally goes to the content of noncommutative algebraic geometry). In the paper I mentioned above, they gave the correct notion for noncommutative subscheme and correct definition for diagonal. Then they defined noncommutative differential operator as certain kind of differential bimodule and developed localization theory for differential operators. Then one can localize differential operators.(Which means (noncommutative)D-modules automatically becomes noncommutative schemes)
Notice
These framework works for any noncommutative ring, either noncommutative D-module(very noncommutative, such as ring of quantized differential operators) or usual commutative D-module(itself noncommutative but almost commutative)