In this article, among other results, we develop a Galois theory of commutative rings under partial actions of finite groups, extending the well-known results by In the celebrated paper by Chase, Harrison and Rosenberg [3] the authors developed a Galois theory for commutative ring extensions S ⊃ R, under the assumptions that S is separable over R, finitely generated and projective as an R-module, and the elements of the Galois group G are pairwise strongly distinct R-automorphisms of S. In particular, Theorem 1.3 of that paper gives several equivalent conditions for the definition of a Galois extension and Theorem 2.3 states a one-to-one correspondence between the subgroups of G and the R-subalgebras of S which are separable and G-strong.On the other hand, partial actions of groups have been introduced in the theory of operator algebras giving powerful tools of their study (see, in particular, [6,7,10,16] and [18]). A related concept, that of a partial representation of a group on a Hilbert space, has been defined independently by Exel [7], and Quigg and Raeburn [18]. Several relevant classes of C * -algebras were deeply investigated in [8-10] from the point of view of partial actions and partial representations of groups, including the Cuntz-Krieger algebras introduced in [4].Given a partial action of a group on an object it is natural to ask whether it is a restriction of a global action defined on a bigger object. Such global action is called a globalization or an enveloping action, provided that certain minimality condition is satisfied which guarantees its uniqueness. Globalizations of partial actions where first considered by F. Abadie in his PhD Thesis of 1999 (see also [1]) and independently by Kellendonk and Lawson in [15]. $ This paper was partially supported by CNPq, CAPES, FAPESP and FAPERGS (Brazil). (M. Dokuchaev), mferrero@mat.ufrgs.br (M. Ferrero), paques@ime.unicamp.br (A. Paques).
In this paper we consider partial actions of groups on algebras and partial skew group rings. After some general results we prove two versions of Maschke's theorem and then we study von Neumann regularity, the prime radical and the Jacobson radical of partial skew group rings. In this way we extend many results which are known for skew group rings.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.