In thinking about various methods of integration, I began to wonder if there was some sort of unifying theory relating integration and ring theory.
For example, would there be a way to make sense of something such as say integrating over a matrix ring? Or integrating over the ring of real quaternions? We of course integrate over rings (viz. $\mathbb{R}, \mathbb{Q,Z}$ (trivial in the Lebesgue sense)) all the time, but I have wondered if there is a way to extend integration ever further to include more types of rings.
Are there examples of methods of integration that include integration over more abstract rings?