Abstract: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 $\otimes$.
Keywords: weak injective module; weak flat module; weak injective dimension; weak flat dimension
DOI: DOI 10.14712/1213-7243.2020.015
AMS Subject Classification: 18G25 16E10 16E30