On the 2-absorbing submodules and zero-divisor graph of equivalence classes of zero divisors
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Let $R$ be a commutative ring, $M$ be a Noetherian $R$-module, and $N$ a 2-absorbing submodule of $M$ such that $r(N :_{R} M)= \mathfrak p$ is a prime ideal of $R$. The main result of the paper states that if $N=Q_1\cap\cdots\cap Q_n$ with $r(Q_i:_RM)=\mathfrak p_i$, for $i=1,\ldots, n$, is a minimal primary decomposition of $N$, then the following statements are true. \begin{itemize} \item[(i)] $\mathfrak p=\mathfrak p_k$ for some $1 \leq k \leq n$. \item[(ii)] For each $j=1,\ldots,n$ there exists $m_j \in M$ such that ${\mathfrak p}_j=(N :_{R} m_{j})$. \item[(iii)] For each $i,j=1,\ldots,n$ either $\mathfrak p_{i} \subseteq \mathfrak p_{j}$ or $\mathfrak p_{j} \subseteq \mathfrak p_{i}$. \end{itemize} Let $\Gamma_E(M)$ denote the zero-divisor graph of equivalence classes of zero divisors of $M$. It is shown that $\{Q_1\cap\cdots\cap Q_{n-1}, Q_1\cap\cdots\cap Q_{n-2},\ldots , Q_1\}$ is an independent subset of $V(\Gamma_E(M))$, whenever the zero submodule of $M$ is a 2-absorbing submodule and $Q_1\cap\cdots\cap Q_n=0$ is its minimal primary decomposition. Furthermore, it is proved that $\Gamma_E(M)[(0 :_{R} M)]$, the induced subgraph of $\Gamma_E(M)$ by $(0 :_{R} M)$, is complete.
Abstract
Let $R$ be a commutative ring, $M$ be a Noetherian $R$-module, and $N$ a 2-absorbing submodule of $M$ such that $r(N :_{R} M)= \mathfrak p$ is a prime ideal of $R$. The main result of the paper states that if $N=Q_1\cap\cdots\cap Q_n$ with $r(Q_i:_RM)=\mathfrak p_i$, for $i=1,\ldots, n$, is a minimal primary decomposition of $N$, then the following statements are true. \begin{itemize} \item[(i)] $\mathfrak p=\mathfrak p_k$ for some $1 \leq k \leq n$. \item[(ii)] For each $j=1,\ldots,n$ there exists $m_j \in M$ such that ${\mathfrak p}_j=(N :_{R} m_{j})$. \item[(iii)] For each $i,j=1,\ldots,n$ either $\mathfrak p_{i} \subseteq \mathfrak p_{j}$ or $\mathfrak p_{j} \subseteq \mathfrak p_{i}$. \end{itemize} Let $\Gamma_E(M)$ denote the zero-divisor graph of equivalence classes of zero divisors of $M$. It is shown that $\{Q_1\cap\cdots\cap Q_{n-1}, Q_1\cap\cdots\cap Q_{n-2},\ldots , Q_1\}$ is an independent subset of $V(\Gamma_E(M))$, whenever the zero submodule of $M$ is a 2-absorbing submodule and $Q_1\cap\cdots\cap Q_n=0$ is its minimal primary decomposition. Furthermore, it is proved that $\Gamma_E(M)[(0 :_{R} M)]$, the induced subgraph of $\Gamma_E(M)$ by $(0 :_{R} M)$, is complete.
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