# 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?

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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-Alvarez 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

You might be interested in Rota-Baxter algebras. They are, among other things, an abstraction of how indefinite integration by parts works. If you are interested, Rota's original paper on the subject is a good read.

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Thanks, Qiaochu, this appears to fit the bill. – mathphysicist Oct 20 '10 at 19:11

If you find the Rota-Baxter algebra viewpoint of interest then an excellent entry point into the literature on differential-algebraic aspects is the the recent work of Guo, e.g. On differential Rota-Baxter algebras and Baxter algebras and differential algebras. See also other papers listed on his home page. The early papers don't focus so much on these aspects so I would not recommend reading them initially.

Also you may find of interest this paper on classification of related operator identities.
Freeman, J. M. On the classification of operator identities. Studies in Appl. Math. 51 (1972), 73-84.

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Thanks, Bill. I'll have a look. – mathphysicist Oct 21 '10 at 15:09