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- Title
Epsilon-strongly graded rings: Azumaya algebras and partial crossed products.
- Authors
Bagio, Dirceu; Martínez, Luis; Pinedo, Héctor
- Abstract
Let G be a group, let A = ⊕ g ∈ G A g be an epsilon-strongly graded ring over G, let R := A 1 be the homogeneous component associated with the identity of G, and let 홿횒회횂 (R) be the Picard semigroup of R. In the first part of this paper, we prove that the isomorphism class [ A g ] is an element of 홿횒회횂 (R) for all g ∈ G . Moreover, the association g ↦ [ A g ] determines a partial representation of G on 홿횒회횂 (R) which induces a partial action γ of G on the center Z (R) of R. Sufficient conditions for A to be an Azumaya R γ -algebra are presented if R is commutative. In the second part, we study when B is a partial crossed product in the following cases: B = M n (A) is the ring of matrices with entries in A, or B = END A (M) = ⊕ l ∈ G Mor A (M , M) l is the direct sum of graded endomorphisms of graded left A-modules M with degree l, or B = END A (M) where M = A ⊗ R N is the induced module of a left R-module N. Assuming that R is semiperfect, we prove that there exists a subring of A which is an epsilon-strongly graded ring over a subgroup of G and it is graded equivalent to a partial crossed product.
- Subjects
RING theory; MATRIX rings; ISOMORPHISM (Mathematics); ENDOMORPHISMS; ENDOMORPHISM rings; ALGEBRA
- Publication
Forum Mathematicum, 2023, Vol 35, Issue 5, p1257
- ISSN
0933-7741
- Publication type
Article
- DOI
10.1515/forum-2022-0262