Abstract
We study modules over the 0-Hecke algebra of the symmetric group. In particular, we focus on submodules of the Bruhat interval modules of Jung, Kim, Lee and Oh, and develop various structural results for these submodules. We derive a formula that describes the simple submodules of any Bruhat interval module, determine the projective covers and injective hulls for certain quotients of the projective indecomposable 0-Hecke modules of Norton that appear as Bruhat interval modules, and translate these results to further modules by employing techniques in category theory. The results obtained shed light on the 0-Hecke modules associated to many noteworthy bases of quasisymmetric functions, and extend various results in the literature.