Partial actions of groups on generalized matrix rings

• Published in: Journal of algebra Vol. 663; pp. 533 - 564
• Main Authors: Bagio, Dirceu, Pinedo, Héctor
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-02-2025
• Subjects: Galois theory; Generalized matrix ring; Morita equivalent partial group actions; Partial action; secondary; Generalized matrix ring; Partial action; Morita equivalent partial group actions; Galois theory; primary
• ISSN: 0021-8693
• DOI: 10.1016/j.jalgebra.2024.09.018

Let n be a positive integer and R=(Mij)1≤i,j≤n be a generalized matrix ring. For each 1≤i,j≤n, let Ii be an ideal of the ring Ri:=Mii and denote Iij=IiMij+MijIj. We give sufficient conditions for the subset I=(Iij)1≤i,j≤n of R to be an ideal of R. Also, suppose that α(i) is a partial action of a group G on Ri, for all 1≤i≤n. We construct, under certain conditions, a partial action γ of G on R such that γ restricted to Ri coincides with α(i). We study the relation between this construction and the notion of Morita equivalent partial group action given in [1]. Moreover, we investigate properties related to Galois theory for the extension Rγ⊂R. Some examples to illustrate the results are considered in the last part of the paper.