We investigate the influence of radiative transport on the growth of the magnetorotational instability (MRI) in accretion discs. A general dispersion relation describing the growth of small disturbances on a homogeneous background shear flow is provided. It includes compressibility and radiative effects in the flux-limited diffusion approximation. All the effects of radiation transport can be subsumed into one single parameter, an effective speed of sound. Radiative diffusion does not affect the region where the MRI operates, but it alters the growth rates of the MRI either by increasing or decreasing them. We report about corresponding numerical simulations, which are also used for a first investigation of the non-linear stage of the MRI in gas-pressure dominated accretion discs with radiation transport included.