We study partial fraction decompositions (PFDs) in several variables using tools from commutative algebra. We give criteria for when a rational function with poles on a hyperplane arrangement has a desirable PFD. Our criteria are obtained by examining the primary decomposition of ideals coming from hyperplane arrangements. We then present an algorithm for finding a PFD that satisfies properties desired for simplifying the calculation of scattering amplitudes. We demonstrate the effectiveness of this algorithm by computing practical examples coming from Feynman integrals.