Squarefree zero-divisor graphs of Stanley-Reisner rings
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Let $\Delta$ be a simplicial complex, $I_\Delta$ its Stanley-Reisner ideal and $K[\Delta]$ its Stanley-Reisner ring over a field $K$. Assume that $\Gamma(R)$ denotes the zero-divisor graph of a commutative ring $R$. Here, first we present a condition on two reduced Noetherian rings $R$ and $R'$, equivalent to $\Gamma(R)\cong \Gamma(R')$. In particular, we show that $\Gamma(K[\Delta]) \cong \Gamma(K'[\Delta'])$ if and only if $|\mathrm{Ass}(I_\Delta)|= |\mathrm{Ass}(I_{\Delta'})|$ and either $|K|,|K'|\leq \aleph_0$ or $|K|=|K'|$. This shows that $\Gamma(K[\Delta])$ contains little information about $K[\Delta]$. Then, we define the squarefree zero-divisor graph of $K[\Delta]$, denoted by $\Gamma_{\mathrm{sf}}(K[\Delta])$, and prove that $\Gamma_{\mathrm{sf}}(K[\Delta]) \cong \Gamma_{\mathrm{sf}} (K[\Delta'])$ if and only if $K[\Delta] \cong K[\Delta']$. Moreover, we show how to find $\dim K[\Delta]$ and $|\mathrm{Ass}(K[\Delta])|$ from $\Gamma_{\mathrm{sf}}(K[\Delta])$.
Abstract
Let $\Delta$ be a simplicial complex, $I_\Delta$ its Stanley-Reisner ideal and $K[\Delta]$ its Stanley-Reisner ring over a field $K$. Assume that $\Gamma(R)$ denotes the zero-divisor graph of a commutative ring $R$. Here, first we present a condition on two reduced Noetherian rings $R$ and $R'$, equivalent to $\Gamma(R)\cong \Gamma(R')$. In particular, we show that $\Gamma(K[\Delta]) \cong \Gamma(K'[\Delta'])$ if and only if $|\mathrm{Ass}(I_\Delta)|= |\mathrm{Ass}(I_{\Delta'})|$ and either $|K|,|K'|\leq \aleph_0$ or $|K|=|K'|$. This shows that $\Gamma(K[\Delta])$ contains little information about $K[\Delta]$. Then, we define the squarefree zero-divisor graph of $K[\Delta]$, denoted by $\Gamma_{\mathrm{sf}}(K[\Delta])$, and prove that $\Gamma_{\mathrm{sf}}(K[\Delta]) \cong \Gamma_{\mathrm{sf}} (K[\Delta'])$ if and only if $K[\Delta] \cong K[\Delta']$. Moreover, we show how to find $\dim K[\Delta]$ and $|\mathrm{Ass}(K[\Delta])|$ from $\Gamma_{\mathrm{sf}}(K[\Delta])$.
참고문헌 (11)
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