  An \textit{algebraic} action of a discrete group $\Gamma $ is a homomorphism
from $\Gamma $ to the group of continuous automorphisms of a compact abelian
group $X$. By duality, such an action of $\Gamma $ is determined by a module
$M=\widehat{X}$ over the integer group ring $\mathbb{Z}\Gamma $ of $\Gamma $.
The simplest examples of such modules are of the form $M=\mathbb{Z}\Gamma
/\mathbb{Z}\Gamma f$ with $f\in \mathbb{Z}\Gamma $; the corresponding algebraic
action is the \textit{principal algebraic $\Gamma $-action} $\alpha _f$ defined
by $f$.
  In this note we prove the following extensions of results by Hayes
\cite{Hayes} on ergodicity of principal algebraic actions: If $\Gamma $ is a
countably infinite discrete group which is not virtually cyclic, and if
$f\in\mathbb{Z}\Gamma $ satisfies that right multiplication by $f$ on $\ell
^2(\Gamma ,\mathbb{R})$ is injective, then the principal $\Gamma $-action
$\alpha _f$ is ergodic (Theorem \ref{t:ergodic2}). If $\Gamma $ contains a
finitely generated subgroup with a single end (e.g. a finitely generated
amenable subgroup which is not virtually cyclic), or an infinite nonamenable
subgroup with vanishing first $\ell ^2$-Betti number (e.g., an infinite
property $T$ subgroup), the injectivity condition on $f$ can be replaced by the
weaker hypothesis that $f$ is not a right zero-divisor in $\mathbb{Z}\Gamma $
(Theorem \ref{t:ergodic1}). Finally, if $\Gamma $ is torsion-free, not
virtually cyclic, and satisfies Linnell's \textit{analytic zero-divisor
conjecture}, then $\alpha _f$ is ergodic for every $f\in \mathbb{Z}\Gamma $
(Remark \ref{r:analytic zero divisor}).
