Preview

Mathematics and Theoretical Computer Science

Advanced search

Some properties of compressed zero-divisor graph

https://doi.org/10.26907/2949-3919.2025.1.52-63

Abstract

We prove that if the compressed zero-divisor graph of a finite associative ring contains only one strong vertex then this vertex has a loop. Moreover, more properties of the compressed zero-divisor graph of a finite associative ring are proved.

About the Author

A. S. Monastyreva
Altai State University
Russian Federation

Anna Sergeevna Monastyreva

61 Lenina av., Barnaul 656049



References

1. S.P. Redmond, The zero-divisor graph of a noncommutative ring, Int. J. Commut. Rings 1 (4), 203–211 (2002).

2. D.F. Anderson, P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (2), 434–447 (1999). DOI: https://doi.org/10.1006/jabr.1998.7840

3. N. Bloomfield, C. Wickham, Local rings with genus two zero divisor graph, Comm. Algebra 38 (8), 2965–2980 (2010). DOI: https://doi.org/10.1080/00927870903100093

4. N. Bloomfield, The zero divisor graphs of commutative local rings of order p4 and p3, Comm. Algebra 41 (2), 765–775 (2013). DOI: https://doi.org/10.1080/00927872.2011.635619

5. E.V. Zhuravlev, A.S. Monastyreva, Compressed zero-divisor graphs of finite associative rings, Siberian Math. J. 61(1), 76-–84 (2020). DOI: https://doi.org/10.1134/S0037446620010061

6. A.S. Monastyreva, The compressed zero-divisor graphs of order 4, J. Algebra Appl. 21(9), art. 2250179 (2022). DOI: https://doi.org/10.1142/S0219498822501791

7. A.A. Afanas’ev, A.S. Monastyreva, Compressed and partially compressed zero-divisor graphs of finite associative rings, Siberian Math. J. 64 (2), 281–291 (2023). DOI: https://doi.org/10.1134/s0037446623020040

8. A.S. Monastyreva, Finite non-nilpotent rings with complete compressed zero-divisor graphs, Lobachevskii J. Math 41 (9), 1666–1671 (2020). DOI: https://doi.org/10.1134/S1995080220090206

9. A.S. Monastyreva, Finite rings with acyclic compressed zero-divisor graphs, Sib. Electron. Math. Rep 21 (1), 405–416 (2024) [in Russian]. DOI: https://doi.org/10.33048/semi.2024.21.030

10. V.P. Elizarov, Finite rings, Gelios ARV, Moscow, 2006 [in Russian].


Review

For citations:


Monastyreva A.S. Some properties of compressed zero-divisor graph. Mathematics and Theoretical Computer Science. 2025;3(1):52-63. (In Russ.) https://doi.org/10.26907/2949-3919.2025.1.52-63

Views: 43


Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.


ISSN 2949-3919 (Online)