Rings over which certain modules are injective

Abstract

This paper is concerned with rings for which all modules in one of the following classes are injective: simple modules, quasi-injective modules, or proper cyclic modules. Such rings are known as V-rings, Ql-rings, and PC-rings, respectively. First, some conditions are developed under which the properties of being a V-ring, QJ-ring, or PCI-ring are left-right symmetric. In the next section, it is shown that a semiprime Goldie ring is a Ql ring if and only if all singular quasi-injective modules are injective. An example is constructed to show that the class of QJ-rings is properly contained in the class of noetherian V-rings. Also, it is shown that the global homological dimension of a QJ-ring cannot be any larger than its Krull dimension. In the final section, it is shown that a V-ring is noetherian if and only if it has a Krull dimension. Examples are put forward to show that a noetherian V-ring may have arbitrary finite Krull dimension.