C. Faith and J. Cozzens have shown that a ring, whose right proper cyclic modules are injective, is either semisimple or a simple, right semihereditary, right Ore V-domain. They have posed a question as to whether such a ring is right noetherian. In this paper, an affirmative answer is given to that question. Moreover, necessary and sufficient conditions are given as to when a right PCI ring is left PCI.