Johns modules and quasi-Johns modules

Abstract

A right Johns ring is a right Noetherian ring in which every right ideal is a right annihilator. It is known that in a Johns ring RR the Jacobson radical J(R)J(R) of RR is nilpotent and Soc(R)(R) is an essential right ideal of RR. Moreover, every right Johns ring RR is right Kasch, that is, every simple right RR-module can be embedded in RR. For a M∈RM∈R-Mod we use the concept of MM-annihilator and define a Johns module (resp. quasi-Johns) as a Noetherian module MM such that every submodule is an MM-annihilator. A module MM is called quasi-Johns if any essential submodule of MM is an MM-annihilator and the set of essential submodules of MM satisfies the ascending chain condition. In this paper we extend classical results on Johns rings, as those mentioned above and we also provide new ones. We investigate when a Johns module is Artinian and we give some information about its prime submodules.