PreprintCC BY-NC-SA 4.0arXiv
Gorenstein homological properties of n-trivial extensions of rings
Abstract
We explicitly investigate Gorenstein projective, injective and flat modules over the n-trivial extension RltimesnM of a ring R by an R-bimodule M. Assume that fd(M⊗RiR)<∞ and pd(RM⊗Ri)<∞ for any 1≤ i≤ n, fd(Z(R)RltimesnM)<∞ and pd(RltimesnMZ(R))<∞. It is proven that a left RltimesnM-module (X,f) is Gorenstein projective if and only if the sequence M⊗Rn+1⊗R Xstackrel(M⊗ f) ⋯(M⊗Rn⊗ f)⟶ M⊗R Xstackrelf⟶ X is exact and coker(f) is a Gorenstein projective left R-module. As a consequence, we characterize Gorenstein projective, injective and flat modules over tensor rings.
Related
- Homological aspects of n-trivial extensions of rings
- Gorenstein homological dimension of group extensions
- On classical and Gorenstein homological invariants of rings
- Relative quasi-Gorenstein homological dimensions in extriangulated categories
- Hochschild cohomology and AS-Gorenstein property of weak Hopf Galois extensions
BibTeX
@misc{mao2026gorenstein,
title = {{Gorenstein homological properties of n-trivial extensions of rings}},
author = {Lixin Mao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36499},
eprint = {2609.36499},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36499},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Gorenstein homological properties of n-trivial extensions of rings
AU - Lixin Mao
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36499
UR - https://arxiv.org/abs/2609.36499
AB - We explicitly investigate Gorenstein projective, injective and flat modules over the n-trivial extension Rltimes_nM of a ring R by an R-bimodule M. Assume that fd(M^⊗_Ri_R)<∞ and pd(_RM^⊗_Ri)<∞ for any 1≤ i≤ n, fd(Z(R)_Rltimes_nM)<∞ and pd(_Rltimes_nMZ(R))<∞. It is proven that a left Rltimes_nM-module (X,f) is Gorenstein projective if and only if the sequence M^⊗_Rn+1⊗_R Xstackrel(M⊗ f) ⋯(M^⊗_Rn⊗ f)⟶ M⊗_R Xstackrelf⟶ X is exact and coker(f) is a Gorenstein projective left R-module. As a consequence, we characterize Gorenstein projective, injective and flat modules over tensor rings.
ER -