On modules with the Kulikov property and pure semisimple modules and rings

For an R-module M let σM] denote the category of submodules of M-generated modules. M has the Kulikov property if submodules of pure projective modules in σM] are pure projective. The following is proved: Assume M is a locally noetherian module with the Kulikov property and there are only finitely many simple modules in σM]. Then, for every n ε , there are only finitely many indecomposable modules of length n in σM].

With our techniques we provide simple proofs for some results on left pure semisimple rings obtained by Prest and Zimmermann-Huisgen and Zimmermann with different methods.