A SIMPLE RING OVER WHICH PROPER CYCLICS ARE CONTINUOUS IS A PCI-RING Page
1 Canad. Math. Bull. Vol. 41 (3), 1998 pp. 261–266 A SIMPLE RING OVER WHICH PROPER …

By a well-known result of Osofsky [6, Theorem] a ring R is semisimple (ie R is right artinian
and the Jacobson radical of R is zero) if and only if every cyclic right R-module is injective …

A ring R is called right PCI if every proper cyclic right R-module is injective, ie if C is a cyclic
right R-module then CR≅ RR or CR is injective. By [2] and [3], if R is a non-artinian right PCI …

Abstract A right R-module C is proper cyclic in case R→ C→ 0 is exact but 0→ R→ C→ 0 is
not. A ring R is a ring PCI ring if every proper cyclic right R-module is injective (PC⇒ I ring”) …

A ring R (with identity element) is called a right PCI-ring if and only if every proper cyclic right
R-module is injective; that is, if C is a cyclic right R-module then either C≌ R or C is …

6 0 GOEL AND JAIN self-injective ring (or if R is a semiperfect ring) then each cyclic R-
module is a-injective if and only if R is a direct sum of semisimple artinian ring and a finite …

A module M over a ring R is called n-injective (also called quasi-continuous) if for every pair
of R-submodules N,, N, of M with N, n N,=(0) each projection n,: N, SN,-N, i= 1, 2, can be …

When cyclic singular modules over a simple ring are injective Page 1 Journal of Algebra 263
(2003) 188–192 www.elsevier.com/locate/jalgebra When cyclic singular modules over a …

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 …

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 …