A right R-module C is propercyclic in case R → C → 0 is exact but 0 → R → C → 0 is not. A ring R is a ringPCI ring if every proper cyclic right R-module is injective (PC ⇒ I ring”). The
paper is devoted to the proof of the THEOREM. A right PCI ring is eithersemisimple (artinian) or a right semihereditary simple right Ore domain. When R is
assumed to be right noetherian, this is a theorem of A. Boyle (Rutgers Ph. D. Thesis
1971). When R is assumed to be right selfinjective, this is a theorem of B.
Osofsky, (Rutgers Ph. D. thesis, 1964) and the proof uses this. Aside from the
theorems of Boyle and Osofsky, interest in right PCI rings stems from Cozzens’s
examples (Rutgers Ph. D. thesis, 1969) of right and left principal ideal PCI
domains. Boyle’s theorems show that noetherian right and left PCI rings
are hereditary. The above theorem shows that in any case PCI rings are
semihereditary. How close they come to being hereditary (resp. principal ideal) rings
is an open question. A counterexample R would have to have an injective
cyclic module R∕I with infinite socle! However, if we assume that R is a free
right ideal ring (fir) then R must be a principal right ideal ring. In any case,
we show for any right PCI ring that any finitely generated right ideal is
generated by two elements. Other resemblances to Dedekind domains are
noted.
