$(m,d)$-Injective Covers and Gorenstein $(m, d)$-Flat Modules
Received:October 23, 2015  Revised:March 18, 2016
Key Words: $(m, d)$-injective cover   Gorenstein $(m, d)$-flat module   Gorenstein $(m, d)$-injecitve module   strongly Gorenstein $(m, d)$-flat module  
Fund Project:Supported by the Provincial Natural Science Research Program of Higher Education Institution of Anhui Province (Grant No.KJ2012Z028).
Author NameAffiliation
Liang ZHAO School of Mathematics and Physics, Anhui University of Technology, Anhui 243032, P. R. China
School of Mathematics Sciences, Nanjing Normal University, Jiangsu 210046, P. R. China 
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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.
Citation:
DOI:10.3770/j.issn:2095-2651.2016.05.004
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