Geodesic
$(n,d)$-injective covers, $n$-coherent rings, and $(n,d)$-rings
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 289-304.

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It is known that a ring $R$ is left Noetherian if and only if every left $R$-module has an injective (pre)cover. We show that $(1)$ if $R$ is a right $n$-coherent ring, then every right $R$-module has an $(n,d)$-injective (pre)cover; $(2)$ if $R$ is a ring such that every $(n,0)$-injective right $R$-module is $n$-pure extending, and if every right $R$-module has an $(n,0)$-injective cover, then $R$ is right $n$-coherent. As applications of these results, we give some characterizations of $(n,d)$-rings, von Neumann regular rings and semisimple rings.
DOI : 10.1007/s10587-014-0101-1
Classification : 16D40, 16D50, 16E40, 16P70, 18G25
Keywords: cover; envelope; $n$-coherent ring; $(n, d)$-injective; $(n, d)$-ring 
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     title = {$(n,d)$-injective covers, $n$-coherent rings, and $(n,d)$-rings},
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Li, Weiqing; Ouyang, Baiyu. $(n,d)$-injective covers, $n$-coherent rings, and $(n,d)$-rings. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 289-304. doi : 10.1007/s10587-014-0101-1. http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0101-1/

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