  Let $A$ be a simple abelian variety of dimension $g$ defined over a finite
field $\mathbb{F}_q$ with Frobenius endomorphism $\pi$. This paper describes
the structure of the group of rational points $A(\mathbb{F}_{q^n})$, for all $n
\geq 1$, as a module over the ring $R$ of endomorphisms which are defined over
$\mathbb{F}_q$, under certain technical conditions. If $[\mathbb{Q}(\pi) :
\mathbb{Q}]=2g$ and $R$ is a Gorenstein ring, then ${A(\mathbb{F}_{q^n}) \cong
R/R(\pi^n-1)}$. This includes the case when $A$ is ordinary and has maximal
real multiplication. Otherwise, if $Z$ is the center of $R$ and $(\pi^n - 1)Z$
is the product of invertible prime ideals in $Z$, then $A(\mathbb{F}_{q^n})^d
\cong R/R(\pi^n - 1)$ where $d = 2g/[\mathbb{Q}(\pi):\mathbb{Q}]$. Finally, we
deduce the structure of $A(\overline{\mathbb{F}}_q)$ as a module over $R$ under
similar conditions. These results generalize results of Lenstra for elliptic
curves.
