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2019 On global $\mathscr C$-dimensions
Weiqing Li, Liang Yan, Baiyu Ouyang
Rocky Mountain J. Math. 49(2): 557-577 (2019). DOI: 10.1216/RMJ-2019-49-2-557

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.

Funding Statement

This research was partially supported by the Natural Science Foundation of Hunan Province, grant Nos. 2018JJ2370 and 2016JJ3095.

Citation

Download Citation

Weiqing Li. Liang Yan. Baiyu Ouyang. "On global $\mathscr C$-dimensions." Rocky Mountain J. Math. 49 (2) 557 - 577, 2019. https://doi.org/10.1216/RMJ-2019-49-2-557

Information

Received: 20 March 2018; Revised: 28 August 2018; Published: 2019
First available in Project Euclid: 23 June 2019

MathSciNet: MR3973240
Digital Object Identifier: 10.1216/RMJ-2019-49-2-557

Subjects:
Primary: 16D50, 16E40, 18G25

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.49 • No. 2 • 2019
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