Actions of Finite-Dimensional Semisimple Hopf Algebras and Invariant Algebras

Commun. Korean Math. Soc. 1998 Vol. 13, No. 2, 225-232

Kang Ju Min, Jun Seok Park Chungnam National University, Hoseo University

Abstract : Let $H$ be a finite dimensional Hopf algebra over a field $k$, and $A$ be an $H$-module algebra over $k$ which the $H$-action on $A$ is $\Cal D$-continuous. We show that $Q_{max}(A)$ , the maximal ring of quotients of $A$, is an $H$-module algebra. This is used to prove that if $H$ is a finite dimensional semisimple Hopf algebra and $A$ is a semiprime right(left) Goldie algebra then $A\#H$ is a semiprime right(left) Goldie algebra. Assume that $A$ is a semiprime $H$-module algebra. Then $A^H$ is left Artinian if and only if $A$ is left Artinian.