The Homology Groups of Zero Divisor Graphs of Finite Commutative Rings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918245280251904 |
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| author | Li, Fenglin |
| author_facet | Li, Fenglin |
| contents | This paper investigates the homology groups of the clique complex associated with the zero-divisor graph of a finite commutative ring. Generalizing the construction introduced by F. R. DeMeyer and L. DeMeyer, we establish a Kunneth-type formula for the homology of such complexes and provide explicit computations for products of finite local rings. As a notable application, we obtain a general method to determine the clique homology groups of Z_n and related ring products. Furthermore, we derive explicit formulas for the Betti numbers when all local factors are fields or non-fields. A complete classification of when this clique complex is Cohen-Macaulay is given, with the exception of one borderline case. Finally, our results yield a partial answer to a question posed in earlier literature, showing that certain topological spaces such as the Klein bottle and the real projective plane cannot be realized as zero-divisor complexes of finite commutative rings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_14224 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Homology Groups of Zero Divisor Graphs of Finite Commutative Rings Li, Fenglin Commutative Algebra Algebraic Topology This paper investigates the homology groups of the clique complex associated with the zero-divisor graph of a finite commutative ring. Generalizing the construction introduced by F. R. DeMeyer and L. DeMeyer, we establish a Kunneth-type formula for the homology of such complexes and provide explicit computations for products of finite local rings. As a notable application, we obtain a general method to determine the clique homology groups of Z_n and related ring products. Furthermore, we derive explicit formulas for the Betti numbers when all local factors are fields or non-fields. A complete classification of when this clique complex is Cohen-Macaulay is given, with the exception of one borderline case. Finally, our results yield a partial answer to a question posed in earlier literature, showing that certain topological spaces such as the Klein bottle and the real projective plane cannot be realized as zero-divisor complexes of finite commutative rings. |
| title | The Homology Groups of Zero Divisor Graphs of Finite Commutative Rings |
| topic | Commutative Algebra Algebraic Topology |
| url | https://arxiv.org/abs/2510.14224 |