Max-injective, max-flat modules and max-coherent rings
Bull. Korean Math. Soc. 2010 Vol. 47, No. 3, 611-622
https://doi.org/10.4134/BKMS.2010.47.3.611
Published online May 1, 2010
Yueming Xiang
Hunan Normal University
Abstract : A ring $R$ is called left max-coherent provided that every maximal left ideal is finitely presented. $\mathscr {MI}$ (resp. $\mathscr {MF}$) 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 $\mathscr{M\!F}$-preenvelope, and every left $R$-module has an $\mathscr{M\!I}$-cover. Furthermore, it is shown that a ring $R$ is left max-injective if and only if any left $R$-module has an epic $\mathscr{M\!I}$-cover if and only if any right $R$-module has a monic $\mathscr{M\!F}$-preenvelope. We also give several equivalent characterizations of $MI$-injectivity and $MI$-flatness. Finally, $\mathscr{M\!I}$-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, $\mathscr{M\!I}$-dimension
MSC numbers : 16E10, 16D50, 16D40
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