2022/08/21 by Dirceu Bagio, Luis Martínez, Bagio, Dirceu +3
Mathematics · #13A50 #16W22 #16W50 #16W55 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2208.09769
openalex publication_date 2022/08/21 · openalex created_date 2022/08/24 · openalex updated_date 2026/07/28
The main purpose of this paper is to investigate epsilon-strongly graded rings that are partial crossed products. Let G be a group, A=⊕g∈ G Ag an epsilon-strongly graded ring and \bf picR the Picard semigroup of R:=A1. We prove that the isomorphism class [Ag] is an element of \bf picR, for all g∈ G. Thus, the association g↦ [Ag] determines a partial representation of G on \bf picR which induces a partial action γ of G on the center Z(R) of R. Sufficient conditions for A to be an Azumaya Rγ-algebra are presented in the case that R is commutative. We study when B is a partial crossed product in the following cases: B=Mn(A) is the ring of matrices with entries in A, or B=\bf grmM=\bigoplusl ∈ G\bf MorA(M,M)l is the direct sum of graded endomorphisms of left graded A-module M with degree l, or B=\bf grmM where M=A⊗RN is the induced module of a left R-module N. Finally, assuming that R is semiperfect, we prove that there exists an epsilon-strongly graded subring of A which is graded equivalent to a partial crossed product.