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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.
References:
[1] Anderson, F. W., Fuller, K. R.: Rings and Categories of Modules. (2nd ed.). Graduate Texts in Mathematics 13 Springer, New York (1992). MR 1245487 | Zbl 0765.16001
[2] Bican, L., Bashir, R. El, Enochs, E.: All modules have flat covers. Bull. Lond. Math. Soc. 33 (2001), 385-390. DOI 10.1017/S0024609301008104 | MR 1832549 | Zbl 1029.16002
[3] Chen, J., Ding, N.: On $n$-coherent rings. Commun. Algebra 24 (1996), 3211-3216. DOI 10.1080/00927879608825742 | MR 1402554 | Zbl 0877.16010
[4] Costa, D. L.: Parameterizing families of non-Noetherian rings. Commun. Algebra 22 (1994), 3997-4011. DOI 10.1080/00927879408825061 | MR 1280104 | Zbl 0814.13010
[5] Damiano, R. F.: Coflat rings and modules. Pac. J. Math. 81 (1979), 349-369. DOI 10.2140/pjm.1979.81.349 | MR 0547604 | Zbl 0415.16021
[6] Ding, N.: On envelopes with the unique mapping property. Commun. Algebra 24 (1996), 1459-1470. DOI 10.1080/00927879608825646 | MR 1380605 | Zbl 0863.16005
[7] Enochs, E. E.: Injective and flat covers, envelopes and resolvents. Isr. J. Math. 39 (1981), 189-209. DOI 10.1007/BF02760849 | MR 0636889 | Zbl 0464.16019
[8] Enochs, E. E., Jenda, O. M. G.: Relative Homological Algebra. De Gruyter Expositions in Mathematics 30 Walter de Gruyter, Berlin (2000). MR 1753146 | Zbl 0952.13001
[9] Facchini, A.: Module Theory: Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules. Progress in Mathematics 167 Birkhäuser, Basel (1998). MR 1634015 | Zbl 0930.16001
[10] Mao, L., Ding, N.: Relative projective modules and relative injective modules. Commun. Algebra 34 (2006), 2403-2418. DOI 10.1080/00927870600649111 | MR 2240382 | Zbl 1104.16002
[11] Pinzon, K.: Absolutely pure covers. Commun. Algebra 36 (2008), 2186-2194. DOI 10.1080/00927870801952694 | MR 2418384 | Zbl 1162.16003
[12] Rotman, J. J.: An Introduction to Homological Algebra. (2nd ed.). Universitext Springer, New York (2009). MR 2455920 | Zbl 1157.18001
[13] Stenström, B.: Coherent rings and $FP$-injective modules. J. Lond. Math. Soc., II. Ser. 2 (1970), 323-329. DOI 10.1112/jlms/s2-2.2.323 | MR 0258888 | Zbl 0194.06602
[14] Xu, J.: Flat Covers of Modules. Lecture Notes in Mathematics 1634 Springer, Berlin (1996). MR 1438789 | Zbl 0860.16002
[15] Ming, R. Yue Chi: On quasi-injectivity and von Neumann regularity. Monatsh. Math. 95 (1983), 25-32. DOI 10.1007/BF01301145 | MR 0697346
[16] Zhou, D.: On $n$-coherent rings and $(n,d)$-rings. Commun. Algebra 32 (2004), 2425-2441. DOI 10.1081/AGB-120037230 | MR 2100480 | Zbl 1089.16001
[17] Zhou, D. X.: Cotorsion pair extensions. Acta Math. Sin., Engl. Ser. 25 (2009), 1567-1582. DOI 10.1007/s10114-009-6385-7 | MR 2544301 | Zbl 1215.16016
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