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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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    @mathphysicist:$I$know you're looking for algebraic counterparts of _indefinite_ integrals, but you may nevertheless be interested in reading about [Daniell integration](http://en.wikipedia.org/wiki/Daniell_integral).2011-09-03

2 Answers 2

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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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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.2010-10-21