The Homology Groups of Zero Divisor Graphs of Finite Commutative Rings

Fenglin Li

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arXiv · 2510.14224

The Homology Groups of Zero Divisor Graphs of Finite Commutative Rings

Fenglin Li

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Abstract

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.

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BibTeX RIS

Fenglin Li. 2025-10-16. The Homology Groups of Zero Divisor Graphs of Finite Commutative Rings. https://arxiv.org/abs/2510.14224

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