# Algebraization of integral calculus

It is well known that the differential calculus has a nice algebraization in terms of the differential rings but what about integral calculus? Of course, one sometimes defines an integral in a differential ring $R$ with a derivation $\partial$ as a projection $\pi: R\rightarrow \tilde R$, where $\tilde R$ is a quotient of $R$ w.r.t. the following equivalence relation: $f\sim g$ iff $f-g$ is in the image of $\partial$, but this is not very intuitive and apparently corresponds to the idea of definite integral over a fixed domain rather than to that of an indefinite one. So my question is:

Are there algebraic counterparts for the concept of an indefinite integral?

• Indefinite integrals are simply solutions of differential equations: g is an indefinite integral of f if g' = f. That's the model used in the theory of integration in finite terms. If that model doesn't work in your context then you need to say more about your motivation. Oct 20, 2010 at 14:21
• @Bill D.: Sure, one can define the integration operator $I$ as the right inverse of the derivation $\partial$ so that $\partial \circ I=\mathrm{id}$ but I just can't help wondering whether there are more "straightforward" ways to handle the matter. Also (although this is somewhat of an idle speculation :)) there may be contexts where you'd like the integration to be the "primary" structure and the derivative to come in second. Oct 20, 2010 at 15:27
• Why don't you find the usual way to deal with this satisfactory? What do you want to achieve? Oct 20, 2010 at 17:06
• Actually, I didn´t know about the algebraization you mention in your original post matphysicist! Any references on the matter you could recommend me? I´m very interested in studying calculus in its algebraic formulation... ...I know this isn´t exactly an answer to your question, so thanks in advance!
– user15483
Sep 2, 2011 at 19:53
• @mathphysicist: I know you're looking for algebraic counterparts of indefinite integrals, but you may nevertheless be interested in reading about Daniell integration. Sep 3, 2011 at 3:52