ABSTRACT
As it is well known a group G acts on a set X if for any g ∈ G there exists a bijection αg of X such α1 = idR and αg ◦αh = αgh, for any g, h ∈ G, where 1 denotes the identity element of G. In the case that R is a ring G acts on R if the above holds for R = X and, in addition, any αg is an automorphism of R, for every g ∈ G. Partial actions of groups appeared independently in various areas of math-
ematics, in particular, in the theory of operator algebras as a powerful tool in their study (see [4], [5], [6], [9], [12]). In the most general setting of a partial action of a group on a set X the definition is as follows:
Definition. Let G be a group with identity element 1 and X be a set. A partial action α of G on X is a collection of subsets Sg ⊆ X (g ∈ G) and bijections αg : Sg−1 → Sg such that
(i) S1 = X and α1 is the identity map of X; (ii) S(gh)−1 ⊇ αh−1(Sh ∩ Sg−1), for any g, h ∈ G; (iii) αg ◦ αh(x) = αgh(x) for any x ∈ αh−1(Sh ∩ Sg−1).
