On rings with one middle class of injectivity domains

Abstract

A module M is said to be modest if the injectivity domain of M is the class of all crumbling modules. In this paper, we investigate the basic properties of modest modules. We provide characterizations of some classes of rings using modest modules. In particular, we show that a ring having the class of crumbling modules as the only right middle class of injectivity domains is either a right V-ring or right Noetherian; and a commutative ring with this property is regular. We also give criteria for a ring having the class of crumbling modules as the only right middle class of injectivity domains.