In general, the integration is defined on a measurable space $(X, \mathcal{A}, \mu)$, where $X$ is the whole space, $\mathcal{A}$ is a $\sigma$-algebra on $X$, and measure $\mu$. Usually integration is defined for real-valued function $f$ or complex-value function, (which is essentially same as real-valued one). Formally, integration on a subset of $E\subset X$ is a function from ring of measurable functions to $\mathbb{R}$: $$ \int_E: f(a) \to \mathbb{R} $$
I am wondering if there is any research or related results on somewhat more general settings. For example, $f$ is a map from $X$ to some Ring $R$ and $\int_E$ is defined to be a map from the ring of $R$-valued functions to Ring $R$.
In order to make it more "useful", it's better to related this definite integration to some formal indefinite integrations such as discussed here
