  Let $G=(V,E)$ be a simple graph. A set $S\subseteq V$ is independent set of
$G$, if no two vertices of $S$ are adjacent. The independence number
$\alpha(G)$ is the size of a maximum independent set in the graph. %An
independent set with cardinality Let $R$ be a commutative ring with nonzero
identity and $I$ an ideal of $R$. The zero-divisor graph of $R$, denoted by
$\Gamma(R)$, is an undirected graph whose vertices are the nonzero
zero-divisors of $R$ and two distinct vertices $x$ and $y$ are adjacent if and
only if $xy = 0$. Also the ideal-based zero-divisor graph of $R$, denoted by
$\Gamma_I(R)$, is the graph which vertices are the set ${x\in R\backslash I |
xy\in I \quad for some \quad y\in R\backslash I\}$ and two distinct vertices
$x$ and $y$ are adjacent if and only if $xy \in I$. In this paper we study the
independent sets and the independence number of $\Gamma(R)$ and $\Gamma_I(R)$.
