When are proper cyclics injective
Carl C. Faith · Pacific Journal of Mathematics · 1973
A right jR-module C is proper cyclic in case R -> C -> 0 is exact but 0 -* .ft-> C -• 0 is not.A ring R is a ring PCI ring if every proper cyclic right i?-module is injective ("PC ==> / ring").The paper is devoted to the proof of the THEOREM.A right PCI ring is either semisimple (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.Introduction* A "good" theory of ring theory is one which