Deriving some properties of Stanley-Reisner rings from their squarefree zero-divisor graphs
Let ΔΔ be a simplicial complex, IΔIΔ its Stanley-Reisner ideal and R=K[Δ]R=K[Δ] its Stanley-Reisner ring over a field KK. In 2018, the author introduced the squarefree zero-divisor graph of RR, denoted by Γsf(R)Γsf(R), and proved that if ΔΔ and Δ′Δ′ are two simplicial complexes, then the graphs Γsf(K[Δ])Γsf(K[Δ]) and Γsf(K[Δ′])Γsf(K[Δ′]) are isomorphic if and only if the rings K[Δ]K[Δ] and K[Δ′]K[Δ′] are isomorphic. Here we derive some algebraic properties of RR using combinatorial properties of Γsf(R)Γsf(R). In particular, we state combinatorial conditions on Γsf(R)Γsf(R) which are necessary or sufficient for RR to be Cohen-Macaulay. Moreover, we investigate when Γsf(R)Γsf(R) is in some well-known classes of graphs and show that in these cases, IΔIΔ has a linear resolution or is componentwise linear. Also we study the diameter and girth of Γsf(R)Γsf(R) and their algebraic interpretations.
Keywords:
Squarefree monomial ideal, simplicial complex, squarefree zero-divisor graph, Cohen-Macaulay ring linear resolution,
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- D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217(2) (1999), 434-447.
- W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, 1996.
- E. Connon and S. Faridi, Chorded complexes and a necessary condition for a monomial ideal to have a linear resolution, J. Combin. Theory Ser. A, 120(7) (2013), 1714-1731.
- F. R. DeMeyer, T. McKenzie and K. Shneider, The zero-divisor graph of a commutative semigroup, Semigroup Forum, 65(2) (2002), 206-214.
- F. DeMeyer and L. DeMeyer, Zero-divisor graphs of semigroups, J. Algebra, 283(1) (2005), 190-198.
- E. Hashemi and R. Amirjan, Zero-divisor graphs of Ore extensions over reversible rings, Canad. Math. Bull., 59(4) (2016), 794-805.
- J. Herzog and T. Hibi, Monomial Ideals, Springer-Verlag, London, 2011.
- D. T. Hoang and T. N. Trung, A characterization of triangle-free Gorenstein graphs and Cohen-Macaulayness of second powers of edge ideals, J. Algebraic Combin., 43(2) (2016), 325-338.
- K. Kimura and N. Terai, Arithmetical rank of Gorenstein squarefree monomial ideals of height three, J. Algebra, 422 (2015), 11-32.
- T. G. Lucas, The diameter of a zero divisor graph, J. Algebra, 301(1) (2006), 174-193.
- A. Nikseresht, Chordality of clutters with vertex decomposable dual and ascent of clutters, J. Combin. Theory Ser. A, 168 (2019), 318-337.
- A. Nikseresht, Squarefree zero-divisor graphs of Stanley-Reisner rings, J. Korean Math. Soc., 55(6) (2018), 1381-1388.
- A. Nikseresht and M. R. Oboudi, Trung's construction and the Charney-Davis conjecture, Bull. Malays. Math. Sci. Soc., 44(1) (2021), 9-16.
- A. Nikseresht and R. Zaare-Nahandi, On generalization of cycles and chordality to clutters from an algebraic viewpoint, Algebra Colloq., 24(4) (2017), 611-624.
- A. Patil, B. N. Waphare and V. Joshi, Perfect zero-divisor graphs, Discrete Math., 340(4) (2017), 740-745.
- S. M. Seyyedi and F. Rahmati, Some algebraic and combinatorial properties of the complete T-partite graphs, Iran. J. Math. Sci. Inform., 13(1) (2018), 131-138.
- D. B. West, Introduction to Graph Theory, 2nd edition, Pearson Education, Singapore, 2001.
- ISSN: 1306-6048
- Yayın Aralığı: Yılda 2 Sayı
- Başlangıç: 2007
- Yayıncı: Abdullah HARMANCI