  We define and study generalizations of simplicial volume over arbitrary
seminormed rings with a focus on $p$-adic simplicial volumes. We investigate
the dependence on the prime and establish homology bounds in terms of $p$-adic
simplicial volumes. As the main examples we compute the weightless and $p$-adic
simplicial volumes of surfaces. This gives a way to calculate classical
simplicial volume of surfaces without hyperbolic straightening and shows that
surfaces satisfy mod $p$ and $p$-adic approximation of simplicial volume.
