### 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 (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 ℐ of natural numbers such that ⊕_{i∈ℐ}, ℰ(S_{i}) is an injective module.

Original language | English |
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Pages (from-to) | 120-123 |

Number of pages | 4 |

Journal | Archiv der Mathematik |

Volume | 78 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2002 Feb 1 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Archiv der Mathematik*,

*78*(2), 120-123. https://doi.org/10.1007/s00013-002-8224-2