Biinvertible actions of Hopf algebras

We study actions of a Hopf algebraH on an algebraR such that the action is twisted by an invertible mapσ:H⊗H →R; the biinvertible condition means that these actions also have both an inverse and an antiinverse in Hom(H, EndR). WhenR is an ordinaryH-module algebra, the action is biinvertible if the antipose is bijective. As a new example we show that if theH-action is twisted and the coradical ofH is cocommutative, then the action is biinvertible. After studying the continuity of these actions with respect to the filter of ideals ofR with zero annihilator, we consider when the actions may be extended to the symmetric Martindale quotient ring ofR and itsH-analog. Our results can be applied to crossed productsR# _σ . Research supported by NSF Grant No. 89-01491.