zbMATH — the first resource for mathematics

Relative weak derived functors. (English) Zbl 07217157
Summary: Let \(R\) be a ring, \(n\) a fixed non-negative integer, \(\mathscr{WI}\) the class of all left \(R\)-modules with weak injective dimension at most \(n\), and \(\mathscr{WF}\) the class of all right \(R\)-modules with weak flat dimension at most \(n\). Using left (right) \(\mathscr{WI}\)-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 \(\mathscr{WF}\times\mathscr{WI}\), and investigate the global right \(\mathscr{WI}\)-dimension of \(_R\mathscr{M}\) by right derived functors of \(\otimes\).
18G25 Relative homological algebra, projective classes (category-theoretic aspects)
16E10 Homological dimension in associative algebras
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
Full Text: DOI
[1] Ding N., On envelopes with the unique mapping property, Comm. Algebra. 24 (1996), no. 4, 1459-1470
[2] Enochs E. E.; Jenda O. M. G., Relative Homological Algebra, De Gruyter Expositions in Mathematics, 30, Walter de Gruyter, Berlin, 2000
[3] Enochs E. E.; Huang Z., Injective envelopes and (Gorenstein) flat covers, Algebr. Represent. Theory 15 (2012), no. 6, 1131-1145
[4] Gao Z.; Wang F., Weak injective and weak flat modules, Comm. Algebra 43 (2015), no. 9, 3857-3868
[5] Gao Z.; Huang Z., Weak injective covers and dimension of modules, Acta Math. Hungar. 147 (2015), no. 1, 135-157
[6] Göbel R.; Trlifaj J., Approximations and Endomorphism Algebra of Modules, De Gruyter Expositions in Mathematics, 41, Walter de Gruyter, Berlin, 2006
[7] Rotman J. J., An Introduction to Homological Algebra, Pure and Applied Mathematics, 85, Academic Press, New York, 1979
[8] Stenström B., Coherent rings and \(FP\)-injective modules, J. London Math. Soc. (2) 2 (1970), 323-329
[9] Xu J., Flat covers of modules, Lecture Notes in Mathematics, 1634, Springer, Berlin, 1996
[10] Zeng Y.; Chen J., Envelopes and covers by modules of finite \(FP\)-injective dimensions, Comm. Algebra. 38 (2010), no. 10, 3851-3867
[11] Zhang D.; Ouyang B., On \(n\)-coherent rings and \((n, d)\)-injective modules, Algebra Colloq. 22 (2015), no. 2, 349-360
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.