# Article

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Keywords:
weak injective module; weak flat module; weak injective dimension; weak flat dimension
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 $\otimes$.
References:
[1] Ding N.: On envelopes with the unique mapping property. Comm. Algebra. 24 (1996), no. 4, 1459–1470. DOI 10.1080/00927879608825646 | MR 1380605
[2] 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
[3] Enochs E. E., Huang Z.: Injective envelopes and (Gorenstein) flat covers. Algebr. Represent. Theory 15 (2012), no. 6, 1131–1145. DOI 10.1007/s10468-011-9282-6 | MR 2994019
[4] Gao Z., Wang F.: Weak injective and weak flat modules. Comm. Algebra 43 (2015), no. 9, 3857–3868. DOI 10.1080/00927872.2014.924128 | MR 3360853
[5] Gao Z., Huang Z.: Weak injective covers and dimension of modules. Acta Math. Hungar. 147 (2015), no. 1, 135–157. DOI 10.1007/s10474-015-0540-7 | MR 3391518
[6] Göbel R., Trlifaj J.: Approximations and Endomorphism Algebra of Modules. De Gruyter Expositions in Mathematics, 41, Walter de Gruyter, Berlin, 2006. MR 2251271
[7] Rotman J. J.: An Introduction to Homological Algebra. Pure and Applied Mathematics, 85, Academic Press, New York, 1979. MR 0538169 | Zbl 1157.18001
[8] Stenström B.: Coherent rings and $FP$-injective modules. J. London Math. Soc. (2) 2 (1970), 323–329. DOI 10.1112/jlms/s2-2.2.323 | MR 0258888
[9] Xu J.: Flat covers of modules. Lecture Notes in Mathematics, 1634, Springer, Berlin, 1996. DOI 10.1007/BFb0094173 | MR 1438789
[10] Zeng Y., Chen J.: Envelopes and covers by modules of finite $FP$-injective dimensions. Comm. Algebra. 38 (2010), no. 10, 3851–3867. DOI 10.1080/00927870903200851 | MR 2760695
[11] Zhang D., Ouyang B.: On $n$-coherent rings and $(n, d)$-injective modules. Algebra Colloq. 22 (2015), no. 2, 349–360. DOI 10.1142/S1005386715000309 | MR 3336067
[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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