Archiv der Mathematik

, Volume 78, Issue 2, pp 120–123

On essential extensions of direct sums of injective modules

  • K. I. Beidar
  • W.-F. Ke

Abstract.

We characterize locally Noetherian modules MR 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.

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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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