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Abstract

A cyclic right R-module M is called proper cyclic if M ≠ R_R A ring R is called a right PCI ring lf every proper cyclic right R-module is injective. Faith (1973a) proved that a right PCI ring is either semisimple (artinian) or a right semihereditary simple right Ore domain. In his introduction, Faith states the reductions to the case R is a domain are long and not entirely satisfactory inasmuch as they are quite intricate. The object of the present paper is to simplify those reductions. We will show that some results in Faiths paper were good enough to yield shorter and dimpler reductions. This will be achieved by first showing that if R is a right PCI ring and if A is a nonzero right ideal of R, then R/A is always injective.