Rings whose quasi-injective modules are injective

Author:
K. A. Byrd

Journal:
Proc. Amer. Math. Soc. **33** (1972), 235-240

MSC:
Primary 16A52

DOI:
https://doi.org/10.1090/S0002-9939-1972-0310009-7

MathSciNet review:
0310009

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Abstract: A ring *R* is called a V-ring, respectively SSI-ring, respectively QII-ring if simple, respectively semisimple, respectively quasi-injective, right *R*-modules are injective. We show that *R* is SSI if and only if *R* is a right noetherian V-ring and that any SSI-ring is a finite ring direct sum of simple SSI-rings. We show that if *R* is left noetherian and SSI then *R* is QII provided *R* is hereditary and that in order for *R* to be hereditary it suffices that maximal right ideals of *R* be reflexive. An example of Cozzens is cited to show these rings need not be artinian.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0310009-7

Keywords:
Quasi-injective module,
V-ring,
Morita-equivalent,
quotient ring

Article copyright:
© Copyright 1972
American Mathematical Society