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MAX-INJECTIVE, MAX-FLAT MODULES AND MAX-COHERENT RINGS

  • Xiang, Yueming (College of Mathematics and Computer Science Hunan Normal University, College of Mathematics and Computer Science Yichun University)
  • Received : 2008.12.21
  • Published : 2010.05.31

Abstract

A ring R is called left max-coherent provided that every maximal left ideal is finitely presented. $\mathfrak{M}\mathfrak{I}$ (resp. $\mathfrak{M}\mathfrak{F}$) denotes the class of all max-injective left R-modules (resp. all max-flat right R-modules). We prove, in this article, that over a left max-coherent ring every right R-module has an $\mathfrak{M}\mathfrak{F}$-preenvelope, and every left R-module has an $\mathfrak{M}\mathfrak{I}$-cover. Furthermore, it is shown that a ring R is left max-injective if and only if any left R-module has an epic $\mathfrak{M}\mathfrak{I}$-cover if and only if any right R-module has a monic $\mathfrak{M}\mathfrak{F}$-preenvelope. We also give several equivalent characterizations of MI-injectivity and MI-flatness. Finally, $\mathfrak{M}\mathfrak{I}$-dimensions of modules and rings are studied in terms of max-injective modules with the left derived functors of Hom.

Keywords

max-injective (pre)cover;max-flat preenvelope;max-coherent ring;MI-injective module;MI-flat module;$\mathfrak{M}\mathfrak{I}$-dimension

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Cited by

  1. Neat-flat Modules vol.44, pp.1, 2016, https://doi.org/10.1080/00927872.2014.982816
  2. Absolutelys-Pure Modules and Neat-Flat Modules vol.43, pp.2, 2015, https://doi.org/10.1080/00927872.2013.842246