Star compressed zero divisors graph and partitions of vector spaces

Let $R$ be a commutaive ring and $Zd(R)$ be the set of zero divisors of $R$. Define an equivalence relation $\sim$ on $Zd(R)$ as follows: $x\sim y$ if and only if $ann(x)=ann(y)$. The graph $\Gamma_E(R)$ is a graph associated to R whose vertices are the classes of elements in $Zd(R)^*=Zd(R)\backslash\{0\}$, and two distinct classes $[x]\neq[y]$ are joined by an edge if and only if $xy=0$. We show that if $R$ is a local ring and $\Gamma_E(R)$ is a star graph with at least four elements then $m/Soc(R)$ has a partition of vector spaces where $m$ is the maximal ideal of $R$. Also, We construct from a special partition of vector spaces, a ring whose associated graph is a star graph.
 
1. Introduction
The compressed zero divisor graph or the graph of equivalence classes of zero divisors of a ring $R$ is denoted by $\Gamma_E(R)$, and is defined in [20]. Let $Zd(R)$ denotes the set of zero divisors of a ring $R$ and $Zd(R)^*=Zd(R)\backslash\{0\}$. Define a relation $\sim$ on $Zd(R)$ as follows [14]: $x\sim y$ if and only if $ann(x)=ann(y)$. It is easily seen that $\sim$ is an equivalence relation. The graph $\Gamma_E(R)$ is a graph associated to R whose vertices are the classes of elements in $Zd(R)^*$, and two distinct classes $[x]\neq[y]$ are joined by an edge if and only if $xy=0$. Another interpretation of $\Gamma_E(R)$ is as follows: The vertices are the elements of $\mathcal{J}=\{ann(a):a\in Zd(R)^*\}$ and two distinct elements $ann(x)$ and $ann(y)$ are adjacent if and only if $xy=0$. In [20], some necessary conditions are obtained for a ring $R$ such that $\Gamma_E(R)$ is a star graph. For example, If $\Gamma_E(R)$ is a star graph with at least four vertices then $|Ass(R)|=1$ and $Char(R)=2,4,8$. In [9] a method for constructing star compressed zero divisor graph is obtained. They used a quotient of a symmetric algebra of a vector space whose relations come from a special partition of that vector space. But it seems that the authors were not aware of partition of vector spaces. By analyzing the proofs in [20] and [9], we see that the star graphs give a partition of a vector space and conversely some partitions give a star graph. An interesting problem about star compressed zero divisor graph is the size of them. For example is there any ring whose compressed zero divisor graph be a star graph with $36$ vertices?
 
2. Main Results
Theorem 2.1. Let $R$ be a ring such that $\Gamma_E(R)$ is a star graph with at least four vertices. Let $[y]$ be the unique vertex with maximal degree and $K=[y]\bigcup \{0\}$. Then $R$ satisfies the following properties:

   (1) $Ass(R)=\{P\}$ where $ann(y)=P$. Also $ann(P)=K$. In particular $K$ is an ideal of $R$ and $Zd(R)=P$.
   (2) $P^3=0$.
   (3) If $ann(x_0)=K$ then $x_0^3=0$ and $[x_0+y]=[x_0]$.
   (4) If $J=\{x\in R:K\subsetneqq ann(x)\}$ then $J$ is an ideal of $R$ and $ann(x)=[x]\bigcup K$ for each $x\in J\backslash K$. Also $[x+y]=[x]$ for each $x\in J\backslash K$.
   (5) $Char(R)=2,4,8$.

Corollary 2.2. Let $(R,m)$ be a local Artinian ring such that $\Gamma_E(R)$ is a star graph with at least four vertices. Let $[y]$ be the unique vertex with maximal degree and $K=[y]\bigcup \{0\}$. If $J=\{x\in R:K\subsetneqq ann(x)\}$ then $\{ann(x)/K:x\in J\backslash (K)\}$ is a partition of $R/m-$vector space $J/K$. Also $Soc(R)=ann(m)=K$.

Theorem 2.3. Let $V=V_1\bigoplus\cdots\bigoplus V_t$ be an $n-$dimensional vector space over $\mathbb{F}_2$ and $dim(V_i)=n_i$. Assume $\{X_{i,k}:1\leq k\leq n_i\}$ is a basis of $V_i$. Let $S=\mathbb{F}_2[X_{i,k}:1\leq i\leq t,1\leq k\leq n_i]$ be a polynomial ring over $n$ indeterminates. Let $I=\langle V_i^2,V^3\rangle$ and $R=\frac{S}{I}$. Then $\Gamma_E(R)$ is a star graph with $2^n-(2^{n_1}+\cdots+2^{n_t})+2t$ vertices.

 
3. Summary of Proofs/Conclusions
In this article we show that every star compressed zero divisor graph correspond to a partition of vector spaces. Conversely, we construct from a special vector space partition a star compressed zero divisor graph.