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A new duality theorem for semisimple modules and characterization of Villamayor rings


Authors: Carl Faith and Pere Menal
Journal: Proc. Amer. Math. Soc. 123 (1995), 1635-1637
MSC: Primary 16E50
DOI: https://doi.org/10.1090/S0002-9939-1995-1254836-6
MathSciNet review: 1254836
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Abstract: We prove the theorem: If R is a ring whose right ideals satisfy the double annihilator condition with respect to a semisimple right R-module W, then every right ideal is an intersection of maximal right ideals, consequently R is a right V (for Villamayor) ring, and W is then necessarily a cogenerator of $ \bmod$-$ R$. (The converse is well known.) We use this to give a new proof of a theorem of ours on right Johns rings.


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

  • [F1] C. Faith, Algebra I: Rings, modules and categories, Grundlehren Math. Wiss., Bd. 190, Springer-Verlag, Berlin, Heidelberg, and New York, 1973 (corrected reprint, 1981). MR 623254 (82g:16001)
  • [F2] -, Minimal cogenerators over Osofsky and Camillo rings, preprint, 1993.
  • [F-M1] C. Faith and Pere Menal, A counter-example to a conjecture of Johns, Proc. Amer. Math. Soc. 116 (1992), 21-26. MR 1100651 (92k:16033)
  • [F-M2] -, The structure of Johns rings, Proc. Amer. Math. Soc. 120 (1994), 1071-1081. MR 1231294 (94j:16036)
  • [P] C. Perelló (ed.), Pere Menal memorial volumes, Publ. Mat. 36 (1992).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1254836-6
Article copyright: © Copyright 1995 American Mathematical Society

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