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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 \(\otimes\).
MSC:
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
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