REGULAR AND BIREGULAR MODULE ALGEBRAS

Motivated by the study of von Neumann regular skew group rings as carried out by Alfaro, Ara, and del RiL.o in [1], 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 between A ^H -Mod and 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 ) (interscction)A ^H using the module theory of A as an module over A ^e - H the enveloping 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.