Max Rings and V-rings

Abstract

This chapter discusses the max rings, V-rings, and rings and modules related to V-rings. All rings are associative and have nonzero identity elements. A module is said to be simple if it does not have nonzero proper submodules. A direct sum of simple modules is called a semi-simple module. A submodule N of a module M is called a maximal submodule if the module M / N is simple. A ring A is called a right max ring if every nonzero right A-module has a maximal submodule. Every semi-simple ring is a right and left max ring, and every factor ring of a right max ring is a right max ring. A ring A is called a right V-ring if every simple right A-module is an injective module. A ring A is called a right GV-ring if all simple singular right A-modules are injective. It is directly verified that A is a right GV-ring as every simple right A-module is either injective or projective.