Regular and Biregular Module Algebras
نویسنده
چکیده
Motivated by the study of von Neumann regular skew groups as carried out by Alfaro, Ara and del Rio 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 is regular and the invariance functor (−) induces an equivalence of A -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 is replaced by R = Z(A)∩A using the module theory of A as a module over A ⊲⊳ 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 Am is H-simple for all maximal ideals m of R. Further sufficient conditions are given for A#H and A to be regular.
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