Gonçalves, Daniel; Öinert, Johan; Royer, Danilo **(2014)**. Simplicity of partial skew group rings with applications to Leavitt path algebras and topological dynamics.* Journal of Algebra, 420,*, 201 - 216

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Authors:

Gonçalves, Daniel
;
Öinert, Johan
;
Royer, Danilo

Department:

Mathematics (Faculty of Engineering)

Algebra

Non-commutative Geometry-lup-obsolete

Algebra

Non-commutative Geometry-lup-obsolete

Research Group:

Algebra

Non-commutative Geometry-lup-obsolete

Non-commutative Geometry-lup-obsolete

Abstract:

Let A be a commutative and associative ring (not necessarily unital), G a group and α a partial action of G on ideals of A, all of which have local units. We show that A is maximal commutative in the partial skew group ring A*G if and only if A has the ideal intersection property in A*G. From this we derive a criterion for simplicity of A*G in terms of maximal commutativity and G-simplicity of A. We also provide two applications of our main results. First, we give a new proof of the simplicity criterion for Leavitt path algebras, as well as a new proof of the Cuntz–Krieger uniqueness theorem. Secondly, we study topological dynamics arising from partial actions on clopen subsets of a compact set.

Let A be a commutative and associative ring (not necessarily unital), G a group and α a partial action of G on ideals of A, all of which have local units. We show that A is maximal commutative in the partial skew group ring A*G if and only if A has the ideal intersection property in A*G. From this we derive a criterion for simplicity of A*G in terms of maximal commutativity and G-simplicity of A. We also provide two applications of our main results. First, we give a new proof of the simplicity criterion for Leavitt path algebras, as well as a new proof of the Cuntz–Krieger uniqueness theorem. Secondly, we study topological dynamics arising from partial actions on clopen subsets of a compact set.

Keywords:

Partial skew group ring ;
Leavitt path algebra ;
Partial topological dynamics ;
Simplicity

ISSN:

0021-8693

LUP-ID:

56a2aaac-d83a-4736-a1ad-efdafeeaf95f | Link: https://lup.lub.lu.se/record/56a2aaac-d83a-4736-a1ad-efdafeeaf95f
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