JOUKNAL OF ALGEBRA 133, 351-372 (1990) opf Galois Extensio an Ben Gurion University, Beersheoa Ml&: Isruai Weizmann Institute, Rehovor 76100, lsrael AND s. ONTGQMERY + University of Southern California. Los Angeles, Cal$wnia 90089 Communicated by the Editors Received June 28. 1959 DEDICATED TO THE MEMORY OF I. N. HERSTEIN, TEACHER AND FRIEND i . INTRQDUCTION In this paper we consider the relationship between an algebra A and its rin invariants Au, under the action of a finite dimensional EL use in an essential way the associated semidirect, or s A # pi of A by H. One of the main a~p~~~at~o~s o H-actions on a division ring D. We prove t equivalent to any of the following: D/DN is an is simple, D is a faithful left D # H-module, 3/ property, or D is isomorphic t conditions, D # Hz Mn(DH). whenever A is an irreducible left A # H-module and A ha Goldie rank. Along the way extension and give several co Galois, but do not seem to be well known. In particular, A/AN is Gaks * The first two authors were supported by the Fund for Basic Research administered by the Israel Academy of Sciences and Humanities. ? The third author was supported by NSF Grant DMS 87-0054i. She also thanks Ben Gurion University, where much of this work was done. for its hospita!ity. : Part of the second author’s contribution is contained in her Ph.D. thesis at Ben Gurioa university, written under the direction of the first autho-. 351 481 :I37 2-x