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 - <p style="text-align: justify;">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&#39;s theorem on the global dimension and its Gorenstein homological version.</p>