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, QI-rings, and PCI-rings, respectively. First, some conditions
are developed under which the properties of being a V -ring, QI-ring, or
PCI-ring are left-right symmetric. In the next section, it is shown that a
semiprime Goldie ring is a QI-ring if and only if all singular quasi-injective
modules are injective. An example is constructed to show that the class
of QI-rings is properly contained in the class of noetherian V -rings. Also,
it is shown that the global homological dimension of a QI-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.
