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Volume 55, Issue 4
$\mathfrak{X}$-Gorenstein Projective Dimensions

Jie Wang, Xiaowei Xu & Zhibing Zhao

DOI: 10.4208/jms.v55n4.22.04

J. Math. Study, 55 (2022), pp. 398-414.

Published online: 2022-11

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  • Abstract

In this paper, we mainly investigate the $\mathfrak{X}$-Gorenstein projective dimension of modules and the (left) $\mathfrak{X}$-Gorenstein global dimension of rings. Some properties of $\mathfrak{X}$-Gorenstein projective dimensions are obtained. Furthermore, we prove that the (left) $\mathfrak{X}$-Gorenstein global dimension of a ring $R$ is equal to the supremum of the set of $\mathfrak{X}$-Gorenstein projective dimensions of all cyclic (left) $R$-modules. This result extends the well-known Auslander's theorem on the global dimension and its Gorenstein homological version.

  • Keywords

Gorenstein projective modules, $\mathfrak{X}$-Gorenstein projective modules, $\mathfrak{X}$-Gorenstein projective dimensions, the Auslander’s theorem.

  • AMS Subject Headings

16D80, 16E10, 16G50

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

1570898331@qq.com (Jie Wang)

xuxw@jlu.edu.cn (Xiaowei Xu)

zbzhao@ahu.edu.cn (Zhibing Zhao)

  • BibTex
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  • TXT
@Article{JMS-55-398, author = {Wang , JieXu , Xiaowei and Zhao , Zhibing}, title = {$\mathfrak{X}$-Gorenstein Projective Dimensions}, journal = {Journal of Mathematical Study}, year = {2022}, volume = {55}, number = {4}, pages = {398--414}, abstract = {

In this paper, we mainly investigate the $\mathfrak{X}$-Gorenstein projective dimension of modules and the (left) $\mathfrak{X}$-Gorenstein global dimension of rings. Some properties of $\mathfrak{X}$-Gorenstein projective dimensions are obtained. Furthermore, we prove that the (left) $\mathfrak{X}$-Gorenstein global dimension of a ring $R$ is equal to the supremum of the set of $\mathfrak{X}$-Gorenstein projective dimensions of all cyclic (left) $R$-modules. This result extends the well-known Auslander's theorem on the global dimension and its Gorenstein homological version.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v55n4.22.04}, url = {http://global-sci.org/intro/article_detail/jms/21161.html} }
TY - JOUR T1 - $\mathfrak{X}$-Gorenstein Projective Dimensions AU - Wang , Jie AU - Xu , Xiaowei AU - Zhao , Zhibing JO - Journal of Mathematical Study VL - 4 SP - 398 EP - 414 PY - 2022 DA - 2022/11 SN - 55 DO - http://doi.org/10.4208/jms.v55n4.22.04 UR - https://global-sci.org/intro/article_detail/jms/21161.html KW - Gorenstein projective modules, $\mathfrak{X}$-Gorenstein projective modules, $\mathfrak{X}$-Gorenstein projective dimensions, the Auslander’s theorem. AB -

In this paper, we mainly investigate the $\mathfrak{X}$-Gorenstein projective dimension of modules and the (left) $\mathfrak{X}$-Gorenstein global dimension of rings. Some properties of $\mathfrak{X}$-Gorenstein projective dimensions are obtained. Furthermore, we prove that the (left) $\mathfrak{X}$-Gorenstein global dimension of a ring $R$ is equal to the supremum of the set of $\mathfrak{X}$-Gorenstein projective dimensions of all cyclic (left) $R$-modules. This result extends the well-known Auslander's theorem on the global dimension and its Gorenstein homological version.

Jie Wang, Xiaowei Xu & Zhibing Zhao. (2022). $\mathfrak{X}$-Gorenstein Projective Dimensions. Journal of Mathematical Study. 55 (4). 398-414. doi:10.4208/jms.v55n4.22.04
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