Vol. 67, No. 2, 1976

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Rings whose proper cyclic modules are quasi-injective

Surender Kumar Jain, Surjeet Singh and Robin Gregory Symonds

Vol. 67 (1976), No. 2, 461–472

A ring R with identity is a right PCQI-ring (PCI-ring) if every cyclic right R-module CR is quasi-injective (injective). Left PCQI-rings (PCI-rings) are similarly defined. Among others the following results are proved: (1) A right PCQI-ring is either prime or semi-perfect. (2) A nonprime nonlocal ring is a right PCQI-ring iff every cyclic right R-module is quasi-injective or R(D   D)
0   D , where D is a division ring. In particular, a nonprime nonlocal right PCQI-ring is also a left PCQI-ring. (3) A local right PCQI-ring with maximal ideal M is a right valuation ring or M2 = (0). (4) A prime local right PCQI-ring is a right valuation domain. (5) A right PCQI-domain is a right Öre-domain. Faith proved (5) for right PCI-domains. If R is commutative then some of the main results of Klatt and Levy on pre-self-injective rings follow as a special case of these results.

Mathematical Subject Classification
Primary: 16A52, 16A52
Received: 19 November 1976
Published: 1 December 1976
Surender Kumar Jain
Surjeet Singh
Robin Gregory Symonds