Bulletin of the
Korean Mathematical Society BKMS

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