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Article

Li, Weiqing ; Ouyang, Baiyu
$(n,d)$-injective covers, $n$-coherent rings, and $(n,d)$-rings. (English). Czechoslovak Mathematical Journal, vol. 64 (2014), issue 2, pp. 289-304
MSC: 16D40, 16D50, 16E40, 16P70, 18G25 | MR 3277736 | Zbl 06391494 | DOI: 10.1007/s10587-014-0101-1
Keywords:
cover; envelope; $n$-coherent ring; $(n,d)$-injective; $(n,d)$-ring
Summary:
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.
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