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Archiv der Mathematik

, Volume 78, Issue 2, pp 120–123 | Cite as

On essential extensions of direct sums of injective modules

  • K. I. Beidar
  • W.-F. Ke

Abstract.

We characterize locally Noetherian modules M R in terms of essential extensions of direct sums of M-injective modules. As a special case (M = R) we obtain that the following conditions are equivalent: (a) R is right Noetherian; (b) any essential extension of the direct sum of any family of injective right R-modules is the direct sum of injective right R-modules; (c) any essential extension of the direct sum of the family \( \{{\cal E}(S_i) \, | \, i = 1, 2, ...\} \) of injective hulls of any family \( \{S_i \, | \, i = 1, 2, ...\} \) of simple right R-modules is the direct sum of injective right R-modules; (d) for any family¶\( \{S_i \, | \, i = 1, 2, ...\} \) of simple right R-modules there exists an infinite subset \( {\cal I} \) of natural numbers such that \( \oplus_{i \in{\cal I}}{\cal E}(S_i) \) is an injective module.

Keywords

Natural Number Injective Module Infinite Subset Essential Extension Injective Hull 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag, Basel 2002

Authors and Affiliations

  • K. I. Beidar
    • 1
  • W.-F. Ke
    • 2
  1. 1.Department of Mathematics, National Cheng Kung University, Tainan, Taiwan,¶ e-mail: beidar@mail.ncku.edu.twTW
  2. 2.Department of Mathematics, National Cheng Kung University, Tainan, Taiwan,¶ e-mail: wfke@mail.ncku.edu.twTW

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