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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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