It is known that the
construction of the ring of fractions S−1A of a commutative ring A by a
multiplicative subset S of A can be extended to the differential case. This means that
for a given differential ring (A,d), the differential ring of fractions of (A,d) by S
is constructed simply by defining a derivation operator on S−1A in terms
of the derivation operator d on A. We seek to explain in the categorical
setting of adiunctions and comonads the reasons for which this and other
constructions can be extended to the differential case. A natural product of this
investigation is the construction of the differential affine scheme of a differential
ring.