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ISSN 1088-6826(online) ISSN 0002-9939(print)



Countable injective modules are sigma injective

Author: Charles Megibben
Journal: Proc. Amer. Math. Soc. 84 (1982), 8-10
MSC: Primary 16A52; Secondary 16A33
MathSciNet review: 633266
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Abstract: In this note we show that a countable injective module is $ \sum $-injective and consequently a ring $ R$ is left noetherian if the category of left $ R$-modules has a countable injective cogenerator. Our proof can be modified to establish the corresponding result for quasi-injective modules. We also give an example of a nonnoetherian commutative ring $ R$ such that the category of $ R$-modules has a countable cogenerator.

References [Enhancements On Off] (What's this?)

  • [1] F. Anderson and K. Fuller, Rings and categories of modules, Springer-Verlag, Berlin, Heidelberg and New York, 1974. MR 0417223 (54:5281)
  • [2] J. Arnold and R. Gilmer, Idempotent ideals and unions of nets of Prüfer domains, J. Sci. Hiroshima Univ. Ser. A-I 31 (1967), 131-145. MR 0227156 (37:2741)
  • [3] A. Cailleau and G. Renault, Étude des modules $ \sum $-quasi-injectifs, C. R. Acad. Sci. Paris A-B 270 (1970), 1391-1394. MR 0327441 (48:5783)
  • [4] C. Faith, Rings with ascending condition on annihilators, Nagoya Math. J. 27 (1966), 179-191. MR 0193107 (33:1328)
  • [5] L. Fuchs, On quasi-injective modules, Scuola Norm. Sup. Pisa 23 (1968), 541-546. MR 0258873 (41:3518)
  • [6] J. Lawrence, A countable self-injective ring is quasi-Frobenius, Proc. Amer. Math. Soc. 65 (1977), 217-220. MR 0442025 (56:414)
  • [7] E. Matlis, Injective modules over noetherian rings, Pacific J. Math. 8 (1958), 511-528. MR 0099360 (20:5800)

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Keywords: Injective module, $ \sum $-injective, cogenerator, noetherian
Article copyright: © Copyright 1982 American Mathematical Society

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