- International Electronic Journal of Algebra
- Vol: 31 Issue: 31
- Deriving some properties of Stanley-Reisner rings from their squarefree zero-divisor graphs
Deriving some properties of Stanley-Reisner rings from their squarefree zero-divisor graphs
Authors : Ashkan Nikseresht
Pages : 121-133
Doi:10.24330/ieja.1058421
View : 14 | Download : 10
Publication Date : 2022-01-17
Article Type : Research
Abstract :Let Δ Δ be a simplicial complex, I Δ IΔ its Stanley-Reisner ideal and R = K [ Δ ] R=K[Δ] its Stanley-Reisner ring over a field K K . In 2018, the author introduced the squarefree zero-divisor graph of R R , denoted by Γ s f ( R ) Γsf(R) , and proved that if Δ Δ and Δ ′ Δ′ are two simplicial complexes, then the graphs Γ s f ( K [ Δ ] ) Γsf(K[Δ]) and Γ s f ( 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 R R using combinatorial properties of Γ s f ( R ) Γsf(R) . In particular, we state combinatorial conditions on Γ s f ( R ) Γsf(R) which are necessary or sufficient for R R to be Cohen-Macaulay. Moreover, we investigate when Γ s f ( R ) Γsf(R) is in some well-known classes of graphs and show that in these cases, I Δ IΔ has a linear resolution or is componentwise linear. Also we study the diameter and girth of Γ s f ( R ) Γsf(R) and their algebraic interpretations.Keywords : Squarefree monomial ideal, simplicial complex, squarefree zero-divisor graph, Cohen-Macaulay ring, linear resolution