Enveloping Actions for Partial Actions on Rings

A partial action is, loosely speaking, an action which is not everywhere-defined. Partial actions on sets have seen widespread study - first the partial actions of groups on sets, then those of monoids, inverse semigroups, inductive groupoids, etc. In the case of partial actions of groups on sets, for example, it is possible to show that any partial action may be 'completed', to give a full action, termed the 'globalisation'.   Also appearing in the literature are the partial actions of groups on rings and (associative linear) algebras. Given such a partial action, people have once again posed the perennial question concerning partial actions: when/how can we construct a full action from a given partial action? The major construction which has emerged in this context is that of the so-called 'enveloping action'. This is an analogue of globalisation for partial actions on rings. However, in contrast to the situation with the globalisation, not every such partial action admits an enveloping action. In this seminar, I will begin by surveying the results concerning the construction of an enveloping action for partial actions of groups on rings. I will then move on to consider generalisations of these results to the monoid case, pointing out certain issues which arise, and presenting two cases where these generalisations work particularly well (though slightly differently): those of inverse monoids and right groups.