# 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. – Bill Dubuque Oct 20 '10 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. – mathphysicist Oct 20 '10 at 15:27
• Why don't you find the usual way to deal with this satisfactory? What do you want to achieve? – Mariano Suárez-Álvarez Oct 20 '10 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 '11 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. – Jesse Madnick Sep 3 '11 at 3:52