On global \(\mathscr{C}\)-dimensions
Assume that $R$ is a ring. Let $\mathscr{C}$ be a class of left $R$-modules and $M$ be a left $R$-module. A homomorphism $\phi: C\rightarrow M$ with $C\in\mathscr{C}$ is called $\mathscr{C}$-precover of $M$ if, for any homomorphism $f:C'\rightarrow M$ with $C'\in \mathscr{C}$, there exists a homomorphism $g:C'\rightarrow C$ such that $\phi g=f$. Moreover, if the only such $g$ are automorphisms of $C$ when $C'=C$ and $f=\phi$, then the $\mathscr{C}$-precover $\phi$ is called a $\mathscr{C}$-cover. Dually, a $\mathscr{C}$-preenvelope and a $\mathscr{C}$-envelope can be defined. We say that $\mathscr{C}$ is a (pre)covering (resp., (pre)enveloping) if every left $R$-module has a $\mathscr{C}$-(pre) cover (resp., $\mathscr{C}$-(pre)envelope). Let $\mathscr{C}$ be a class of left $R$-modules and $M$ a left $R$-module. A left $\mathscr{C}$-resolution of $M$ means a $\text{Hom}(\mathscr{C}, -)$ exact complex $\cdots\rightarrow C_1\rightarrow C_0\rightarrow M\rightarrow 0$ (not necessarily exact) with each $C_i\in\mathscr{C}$. Similarly, a right $\mathscr{C}$-resolution of $M$ means a $\text{Hom}(-, \mathscr{C})$ exact complex $0\rightarrow M\rightarrow C^0\rightarrow C^1\rightarrow\cdots $(not necessarily exact) with each $C^i\in\mathscr{C}$. Assume that $\mathscr{C}$ is a precovering. The left $\mathscr{C}$-dimension of $M$, left $\mathscr{C}$-$\dim M$, is defined to be the least positive integer $m$ such that there exists a left $\mathscr{C}$-resolution $0\rightarrow C_m\rightarrow\dots\rightarrow C_1\rightarrow C_0\rightarrow M\rightarrow 0$ of $M$. If there is no such $m$, we say left $\mathscr{C}$-$\dim M=\infty$. The global left $\mathscr{C}$-dimension of $\mathcal{M}$ (where $\mathcal{M}$ is the category of left $R$-modules), $\text{gl left }\mathscr{C}$-$\dim \mathcal{M}$ is defined as $\sup\{\text{left }\mathscr{C}$-$\dim M\mid M\in\mathcal{M}\}$ and is infinite otherwise. The right $\mathscr{C}$-$\dim M$ and the global right $\mathscr{C}$-dimension of $\mathcal{M}$ are defiend similarly. See [\textit{E. E. Enochs} and \textit{O. M. G. Jenda}, Relative homological algebra. Berlin: Walter de Gruyter (2000; Zbl 0952.13001)] for more details on these materials. When $\mathscr{C}$ is a class of left $R$-modules, global $\mathscr{C}$-dimenions of $\mathcal{M}$ are mainly studied. In particular, if every nonzero left $R$-module has a nonzero $\mathscr{C}$-preenvelope and a nonzero $\mathscr{C}$-precover, then it is shown that gl left $\mathscr{C}$-$\dim \mathcal{M} = \text{gl right }\mathscr{C}$-$\dim \mathcal{M}$.
scientific article; zbMATH DE number 7079984
scientific article