19-25 June 2011

Bled, Slovenia

Europe/Ljubljana timezone

# ZERO-DIVISOR GRAPH WITH RESPECT TO IDEALS IN COMMUTATIVE RINGS

Presented by Dr. Ahmad YOUSEFIAN DARANI

Type: Oral presentation

Track: General session

## Content

Let $R$ be a commutative ring. The idea of associating a graph with the zero-divisors $R$ was introduced by Beck in 1988, where he talked about the colorings of such graphs. By the definition he gave, every element of the ring $R$ was a vertex in the graph, and two vertices $x, y$ were adjacent if and only if $x y = 0$ (\cite{B}). Later, D. F. Anderson and P. S. Livingston (\cite{AL}) considered only non-zero zero-divisors as vertices of the graph.
Redmond \cite{R} introduced and studied the definition of the zero-divisor graph with respect to an ideal. Let $I$ be an ideal of a ring $R$. The zero-divisor graph of $R$ with respect to $I$ is an undirected graph, denoted by $\Gamma_{I} (R)$, with vertices $\{ x \in R\backslash I: x y \in I \, \, \mbox{for some} \, \, y \in R\backslash I \}$ where distinct vertices $x$ and $y$ are adjacent if and only if $x y \in I$. Therefore, if $I = 0$ then $\Gamma_{I}(R) = \Gamma (R)$. It is shown in \cite{R} that, $\Gamma_I(R)$ is connected with $diam(\Gamma_I(R))\leq 3$ and if $\Gamma_I(R)$ contains a cycle, then the $gr(\Gamma_I(R))$ is $3, 4$ or $\infty$.
Let $I$ be an ideal of $R$. $I$ is called a radical ideal if $I = \sqrt{I}$, where $\sqrt{I}$ is the set of all elements of $a$ of $R$ with $a^n\in I$ for some positive integer $n$. $I$ is called quasi-primary if $\sqrt{I}$ is a prime ideal of $R$. Also we denote by $\overline{a}$, the coset $a+I$ in $R/I$. We say that $R$ is a chained ring if the ideals of $R$ are linearly ordered by inclusion. It is easy to see that $R$ is a chained ring if and only if eithet $x|y$ or $y|x$ for all $x, y\in R$.
We recall from \cite{F}, that an element $a \in R$ is called prime to an ideal $I$ of $R$ if $r a \in I$ (where $r \in R$) implies that $r\in I$. Denote by $S(I)$ the set of all elements of $R$ that are not prime to $I$. A proper ideal $I$ of $R$ is said to be primal if $S(I)$ forms an ideal;
this ideal is always a prime ideal, called the adjoint ideal $P$ of $I$. In this case we also say that $I$ is a $P$-primal ideal of $R$.
In this talk we give some new results concerning the ideal-based zero-divisor graph. We prove that if $I$ is a primary ideal, then $diam(\Gamma_I(R))\leq 2$. We show also that if $I$ is an ideal of $R$ with $S(I)^2\nsubseteq I$ such that the prime ideals of $R$ contained in $S(I)$ are linearly ordered, then $diam(\Gamma_I(R)) = 2$. Next we show that if $I$ is an ideal of $R$ such that $R/I$ is a chained ring, then $diam(\Gamma_I(R))\leq 2$. Finally, we show that if $I$ is a quasi-primary ideal of $R$ with $\sqrt{I}\subsetneq S(I)$ and $|\sqrt{I}\backslash I|\geq 2$, then $gr(\Gamma_I(R)) = 3$ or $\infty$.
\begin{thebibliography}{99999}
\bibitem{AL} D. F. Anderson and P. S. Livingston,
{\it The zero-divisor graph of a commutative ring},
J. Algebra {\bf 217} (1999), 434--447.
\bibitem{B} I. Beck,
{\it Coloring of commutative rings}.
J. Algebra, {\bf 116} (1988), 208--226.
\bibitem{F} L. Fuchs,
{\it On primal ideals},
Proc. Amer. Math. Soc. {\bf 1} (1950), 1--6.
\bibitem{R} S. P. Redmond,
{\it An ideal-based zero-divisor graph of a commutative ring},
Comm. Algebra {\bf 31} (2003), 4425--4443.
\end{thebibliography}

## Place

Location: Bled, Slovenia

Address: Best Western Hotel Kompas Bled

## Primary authors

- Dr. Ahmad YOUSEFIAN DARANI Assistant Professor, University of Mohaghegh Ardabili