$(m,d)$-Injective Covers and Gorenstein $(m, d)$-Flat Modules

Abstract:

      We consider the conditions under which the class of $(m, d)$-injective $R$-modules is (pre)covering. It is shown that every left $R$-module over a left $(m, d)$-coherent ring has an $(m, d)$-injective cover. Moreover, the classes of Gorenstein $(m, d)$-flat modules and Gorenstein $(m, d)$-injecitve modules are introduced and studied. For a right $(m, d)$-coherent ring $R$, we prove that a left $R$-module $M$ is Gorenstein $(m, d)$-flat if and only if $M^{+}$ is Gorenstein $(m, d)$-injective as a right $R$-module. Some results on Gorenstein flat modules and Gorenstein $n$-flat modules are generalized.