On global $\mathscr C$-dimensions

Abstract

Let $R$ be a ring and $\mathcal {M}$ the left or right $R$-module category. Let $\mathscr {C}$ be a class of $R$-modules, closed under extensions, finite direct sums, direct summands and isomorphisms. Suppose that $\mathscr {C}$ is both precovering and preenveloping. We prove that every $m$th $\mathscr {C}$-cosyzygy of any $R$-module is contained in $\mathscr {C}$ if and only if every $m$th $\mathscr {C}$-syzygy of any $R$-module is contained in $\mathscr {C}$; if $\mathscr {C}$ is closed under cokernels of monomorphisms, then gl right $\mathscr {C}$-${\dim }\, {\mathcal {M}}\leq m$ and every $m$th and $(m+1)$th $\mathscr {C}$-cosyzygy of any $R$-module have a monic $\mathscr {C}$-preenvelope if and only if gl left $\mathscr {C}$-${\dim }\, {\mathcal {M}} \leq m-2$, $m\geq 2$; if $\mathscr {C}$ is closed under kernels of epimorphisms, then gl left $\mathscr {C}$-${\dim }\, {\mathcal {M}}\leq m$ and every $m$th and $(m+1)$th $\mathscr {C}$-syzygy of any $R$-module have an epic $\mathscr {C}$-precover if and only if gl right $\mathscr {C}$-${\dim }\,{\mathcal {M}}\leq m-2$, $m\geq 2$; if every nonzero $R$-module has a nonzero $\mathscr {C}$-preenvelope and a nonzero $\mathscr {C}$-precover, then gl right $\mathscr {C}$-${\dim }\, {\mathcal {M}} =$ gl left $\mathscr {C}$-${\dim }\, {\mathcal {M}}$. Some applications are given. Some known results are extended or improved.