Vol. 58, No. 1, 1975

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Rings over which certain modules are injective

Ann K. Boyle and Kenneth R. Goodearl

Vol. 58 (1975), No. 1, 43–53

This paper is concerned with rings for which all modules in one of the following classes are injective: simple modules, quasi-injective modules, or proper cyclic modules. Such rings are known as V -rings, QI-rings, and PCI-rings, respectively. First, some conditions are developed under which the properties of being a V -ring, QI-ring, or PCI-ring are left-right symmetric. In the next section, it is shown that a semiprime Goldie ring is a QI-ring if and only if all singular quasi-injective modules are injective. An example is constructed to show that the class of QI-rings is properly contained in the class of noetherian V -rings. Also, it is shown that the global homological dimension of a QI-ring cannot be any larger than its Krull dimension. In the final section, it is shown that a V -ring is noetherian if and only if it has a Krull dimension. Examples are put forward to show that a noetherian V -ring may have arbitrary finite Krull dimension.

Mathematical Subject Classification
Primary: 16A52
Received: 30 April 1974
Published: 1 May 1975
Ann K. Boyle
Kenneth R. Goodearl
University of California, Santa Barbara
Santa Barbara CA
United States