On the induced partial action of a quotient group and a structure theorem for a partial Galois extension

Abstract

Let [Formula: see text] be a ring with a partial action [Formula: see text] of a finite group [Formula: see text]. We determine when a quotient group of [Formula: see text] gives rise to a partial action induced by [Formula: see text] on a subring of [Formula: see text]. As an application, we show that if [Formula: see text] is an [Formula: see text]-partial Galois extension and [Formula: see text] is a normal subgroup of [Formula: see text], then under certain conditions [Formula: see text] under the partial action of [Formula: see text] induced by [Formula: see text], denoted by [Formula: see text], is a partial Galois extension. This result was just shown recently by Bagio et al., applying globalization to derive a partial action of [Formula: see text] on [Formula: see text], totally different from the one presented in this paper arising from a Boolean ring generated by certain idempotents. We further show in either type of induced partial action of a quotient group that under certain conditions [Formula: see text] is a DeMeyer–Kanzaki [Formula: see text]-partial Galois extension whenever [Formula: see text] is an [Formula: see text]-partial Galois Azumaya extension. A structure theorem for a partial Galois extension is also presented, namely, every partial Galois extension can be decomposed as a direct sum of Galois extensions and possibly a trivial partial Galois extension of type II.