A ring R with identity
is a right PCQI-ring (PCI-ring) if every cyclic right R-module C≇R is
quasi-injective (injective). Left PCQI-rings (PCI-rings) are similarly defined.
Among others the following results are proved: (1) A right PCQI-ring is either
prime or semi-perfect. (2) A nonprime nonlocal ring is a right PCQI-ring iff
every cyclic right R-module is quasi-injective or R≅ , where D is a
division ring. In particular, a nonprime nonlocal right PCQI-ring is also a left
PCQI-ring. (3) A local right PCQI-ring with maximal ideal M is a right valuation
ring or M2= (0). (4) A prime local right PCQI-ring is a right valuation
domain. (5) A right PCQI-domain is a right Öre-domain. Faith proved (5) for
right PCI-domains. If R is commutative then some of the main results of
Klatt and Levy on pre-self-injective rings follow as a special case of these
results.
, where 