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A ring R is called left max-coherent provided that every maximal left ideal is finitely presented. MI (resp. 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 MF-preenvelope, and every left R-module has an MIcover. Furthermore, it is shown that a ring R is left max-injective if and only if any left R-module has an epic MI-cover if and only if any right R-module has a monic MF-preenvelope. We also give several equivalent characterizations of MI-injectivity and MI-flatness. Finally, MI dimensions of modules and rings are studied in terms of max-injective modules with the left derived functors of Hom. © 2010 The Korean Mathematical Society.

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