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We define the twisted de Rham cohomology and show how to use it to define thenotion of an integral of the form $\int g(x) e^{f(x)}dx$ over an arbitraryring. We discuss also a definition of a family of integrals and some propertiesof the homological definition of integral. We show how to use the twisted deRham cohomology in order to define the Frobenius map on the p-adic cohomology.Finally, we consider two-dimensional topological quantum field theories withgeneral coefficients.

Twisted de Rham cohomology, homological definition of the integral and "Physics over a ring"