A ring R is called a right
PCI-ring if and only if for any cyclic right R-module C either C≅R or C is
injective. Faith has shown that right PCI-rings are either semiprime Artinian or
simple right semihereditary right Ore domains. Thus if R1 and R2 are right
PCI-rings then R = R1⊕ R2 is not a right PCI-ring unless R1 and R2 are
both semiprime Artinian but R has the property that every cyclic right
R-module is the direct sum of a projective right R-module and an injective right
R-module, and rings with this property on cyclic right R-modules will be called
right CDPI-rings. On the other hand, if S is a semiprime Artinian ring then
the ring of 2 × 2 upper triangular matrices with entries in S is also a right
CDPI-ring. The structure of right Noetherian right CDPI-rings is discussed.
These rings are finite direct sums of right Artinian rings and simple rings.
A classification of right Artinian right CDPI-rings is given. However the
structure of simple right Noetherian right CDPI-rings is more difficult to
determine precisely and the problem of finding it reduces to a conjecture of
Faith.
