Partial actions and quotient rings

Jesús Ávila

doi:10.4067/s0716-09172011000200006

Abstract

In this paper we study the Martindale ring of α-quotients Q associated with the partial action (R,α). Among other results we extend the partial action to Q and prove that it can be identified with an ideal of Q, the Martindale ring of β-quotients of T , where (T, β) denotes the enveloping action of (R,α). We prove that, in general, (Q, β) is not the enveloping action of (Q,α) and study the relationship between the rings R, Q, T and Q. Finally, we establish some properties related to the center of Q and the extended α-centroid of R.

Highlights

Partial actions of groups have been considered in many contexts
They are a powerful tool in the generalization of known results of global actions in several areas as partial Galois theory, skew polynomial rings, skew group rings, fixed rings, Hopf algebras and entwining structures
These rings of quotients associated with semiprime rings have since proved to be useful for the theory of rings with polynomial identities, and for the Galois theory of noncommutative rings and for the study of prime ideals under ring extensions in general

Summary

Introduction

Partial actions of groups have been considered in many contexts. This theory were introduced in the theory of operator algebras (see [6], [7] and the literature quoted therein). The partial actions on algebras in a purely algebraic framework were introduced by Dokuchaev and Exel in [6] They are a powerful tool in the generalization of known results of global actions in several areas as partial Galois theory, skew polynomial rings, skew group rings, fixed rings, Hopf algebras and entwining structures. The generalization to semiprime rings was due to Amitsur These rings of quotients associated with semiprime rings have since proved to be useful for the theory of rings with polynomial identities, and for the Galois theory of noncommutative rings and for the study of prime ideals under ring extensions in general. Ferrero ([7]) proved that any proper partial action α on a semiprime ring R possesses a weak enveloping action.

Preliminaries

Other properties