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- Title
On the induced partial action of a quotient group and a structure theorem for a partial Galois extension.
- Authors
Kuo, Jung-Miao; Szeto, George
- Abstract
Let S be a ring with a partial action α of a finite group G. We determine when a quotient group of G gives rise to a partial action induced by α on a subring of S. As an application, we show that if S / S α is an α -partial Galois extension and K is a normal subgroup of G , then under certain conditions S α K / S α under the partial action of G / K induced by α , denoted by α G / K , is a partial Galois extension. This result was just shown recently by Bagio et al., applying globalization to derive a partial action of G / K on S α K , 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 S α K / S α is a DeMeyer–Kanzaki α G / K -partial Galois extension whenever S / S α is an α -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.
- Subjects
QUOTIENT rings; FINITE groups; IDEMPOTENTS; CONTINUED fractions
- Publication
International Journal of Algebra & Computation, 2023, Vol 33, Issue 5, p989
- ISSN
0218-1967
- Publication type
Article
- DOI
10.1142/S0218196723500431