  Motivated by the study of von Neumann regular skew groups as carried out by
Alfaro, Ara and del Rio in 1995 we investigate regular and biregular Hopf
module algebras. If $A$ is an algebra with an action by an affine Hopf algebra
$H$, then any $H$-stable left ideal of $A$ is a direct summand if and only if
$A^H$ is regular and the invariance functor $(-)^H$ induces an equivalence of
$A^H$-Mod to the Wisbauer category of $A$ as $A# H$-module. Analogously we show
a similar statement for the biregularity of $A$ relative to $H$ where $A^H$ is
replaced by $R=Z(A)\cap A^H$ using the module theory of $A$ as a module over
$A\otimes A^{op} \bowtie H$ the envelopping Hopf algebroid of $A$ and $H$. We
show that every two-sided $H$-stable ideal of $A$ is generated by a central
$H$-invariant idempotent if and only if $R$ is regular and $A_m$ is $H$-simple
for all maximal ideals $m$ of $R$. Further sufficient conditions are given for
$A# H$ and $A^H$ to be regular.
