Some remarks on multiplication and projective modules

The associated ideal θ(M) and the trace ideal T(M) of a module M play analogous but distinct roles in the study of multiplication and projective modules respectively. We further investigate both and show in particular that faithful multiplication and projective modules have common properties. We use a method based on Anderson's characterizations of multiplication modules which enables new proofs of theorems on multiplication modules of El Bast, Low, Smith and Smith, and then, also enables similar results for projective modules.