# Article

 Title: Relative weak derived functors (English) Author: Prabakaran, Panneerselvam Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 61 Issue: 1 Year: 2020 Pages: 35-50 Summary lang: English . Category: math . Summary: Let $R$ be a ring, $n$ a fixed non-negative integer, ${\mathscr{W I}}$ the class of all left $R$-modules with weak injective dimension at most $n$, and ${\mathscr{W F}}$ the class of all right $R$-modules with weak flat dimension at most $n$. Using left (right) ${\mathscr{W I}}$-resolutions and the left derived functors of Hom we study the weak injective dimensions of modules and rings. Also we prove that $- \otimes -$ is right balanced on ${\mathscr{M}}_R \times {_R{\mathscr{M}}}$ by ${\mathscr{W F}} \times {\mathscr{W I}}$, and investigate the global right ${\mathscr{W I}}$-dimension of $_R{\mathscr{M}}$ by right derived functors of $\otimes$. (English) Keyword: weak injective module Keyword: weak flat module Keyword: weak injective dimension Keyword: weak flat dimension MSC: 16E10 MSC: 16E30 MSC: 18G25 idZBL: Zbl 07217157 idMR: MR4093428 DOI: 10.14712/1213-7243.2020.015 . Date available: 2020-04-30T11:15:02Z Last updated: 2020-08-26 Stable URL: http://hdl.handle.net/10338.dmlcz/148074 . Reference: [1] Ding N.: On envelopes with the unique mapping property.Comm. Algebra. 24 (1996), no. 4, 1459–1470. MR 1380605, 10.1080/00927879608825646 Reference: [2] Enochs E. E., Jenda O. M. G.: Relative Homological Algebra.De Gruyter Expositions in Mathematics, 30, Walter de Gruyter, Berlin, 2000. Zbl 0952.13001, MR 1753146 Reference: [3] Enochs E. E., Huang Z.: Injective envelopes and (Gorenstein) flat covers.Algebr. Represent. Theory 15 (2012), no. 6, 1131–1145. MR 2994019, 10.1007/s10468-011-9282-6 Reference: [4] Gao Z., Wang F.: Weak injective and weak flat modules.Comm. Algebra 43 (2015), no. 9, 3857–3868. MR 3360853, 10.1080/00927872.2014.924128 Reference: [5] Gao Z., Huang Z.: Weak injective covers and dimension of modules.Acta Math. Hungar. 147 (2015), no. 1, 135–157. MR 3391518, 10.1007/s10474-015-0540-7 Reference: [6] Göbel R., Trlifaj J.: Approximations and Endomorphism Algebra of Modules.De Gruyter Expositions in Mathematics, 41, Walter de Gruyter, Berlin, 2006. MR 2251271 Reference: [7] Rotman J. J.: An Introduction to Homological Algebra.Pure and Applied Mathematics, 85, Academic Press, New York, 1979. Zbl 1157.18001, MR 0538169 Reference: [8] Stenström B.: Coherent rings and $FP$-injective modules.J. London Math. Soc. (2) 2 (1970), 323–329. MR 0258888, 10.1112/jlms/s2-2.2.323 Reference: [9] Xu J.: Flat covers of modules.Lecture Notes in Mathematics, 1634, Springer, Berlin, 1996. MR 1438789, 10.1007/BFb0094173 Reference: [10] Zeng Y., Chen J.: Envelopes and covers by modules of finite $FP$-injective dimensions.Comm. Algebra. 38 (2010), no. 10, 3851–3867. MR 2760695, 10.1080/00927870903200851 Reference: [11] Zhang D., Ouyang B.: On $n$-coherent rings and $(n, d)$-injective modules.Algebra Colloq. 22 (2015), no. 2, 349–360. MR 3336067, 10.1142/S1005386715000309 Reference: [12] Zhao T.: Homological properties of modules with finite weak injective and weak flat dimensions.Bull. Malays. Math. Sci. Soc. 41 (2018), no. 2, 779–805. MR 3781545 .

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