Affiliation:
1. Departamento de Matemática , Universidade Federal de Santa Catarina , 88040-900 Florianópolis , Brazil
2. Department of Mathematics and Natural Sciences , Blekinge Institute of Technology , Karlskrona , Sweden
Abstract
Abstract
Given a partial action α of a groupoid G on a ring R, we study the associated partial skew groupoid ring
R
⋊
α
G
{R\rtimes_{\alpha}G}
,
which carries a natural G-grading.
We show that there is a one-to-one correspondence between the G-invariant ideals of R and the graded ideals of the G-graded ring
R
⋊
α
G
{R\rtimes_{\alpha}G}
. We provide sufficient conditions for primeness, and necessary and sufficient conditions for simplicity of
R
⋊
α
G
{R\rtimes_{\alpha}G}
. We show that every ideal of
R
⋊
α
G
{R\rtimes_{\alpha}G}
is graded if and only if α has the residual intersection property. Furthermore, if α is induced by a topological partial action θ, then we prove that minimality of θ is equivalent to G-simplicity of R, topological transitivity of θ is equivalent to G-primeness of R, and topological freeness of θ on every closed invariant subset of the underlying topological space is equivalent to α having the residual intersection property. As an application, we characterize condition (K) for an ultragraph in terms of topological properties of the associated partial action and in terms of algebraic properties of the associated ultragraph algebra.
Funder
Fundação de Amparo à Pesquisa e Inovação do Estado de Santa Catarina
Conselho Nacional de Desenvolvimento Científico e Tecnológico
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Subject
Applied Mathematics,General Mathematics
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Ring theoretic properties of partial skew groupoid rings with applications to Leavitt path algebras,
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