Rings whose modules satisfy chain conditions on essential endosubmodules

We study rings R such that all (or certain classes) of right R-modules satisfy chain conditions on essential endosubmodules. It is proved that if all right R-modules have DCC on essential endosubmodules, then R is a right pure semisimple ring, and if all right R-modules have both ACC and DCC on essential endosubmodules, then R is of finite representation type. Rings R such that all right R-modules have ACC on essential endosubmodules are discussed, and it is shown, in particular, that if such a ring R is right Artinian, then R is left pure semisimple.