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 $R$ the Jacobson radical $J\left(R\right)$ of $R$ is nilpotent and Soc$\left(R\right)$ is an essential right ideal of $R$. Moreover, every right Johns ring $R$ is right Kasch, that is, every simple right $R$-module can be embedded in $R$. For a $M\in R$-Mod we use the concept of $M$-annihilator and define a Johns module (resp. quasi-Johns) as a Noetherian module $M$ such that every submodule is an $M$-annihilator. A module $M$ is called quasi-Johns if any essential submodule of $M$ is an $M$-annihilator and the set of essential submodules of $M$ 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.

Keywords

• Johns module
• Noetherian module
• M-annihlator
• Kasch module
• (semi)prime submodule
• projective module

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