Local rings of exchange rings

Ara, Pere; Lozano, Miguel Gómez; Molina, Mercedes Siles

doi:10.1080/00927879808826405

Cited by 22 publications

(10 citation statements)

References 11 publications

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“…Remark 2.5. Notice that, according to Theorem 3.5 in [4], every ring has a largest exchange ideal with respect to the inclusion (which might be the zero ideal). Proof: Clearly, if K 0 (I) = 0, then the connecting map δ vanishes and hence Theorem 2.4 applies.…”

Section: Preliminary Resultsmentioning

confidence: 99%

Lifting units modulo exchange ideals and C* -algebras with real rank zero

Perera¹

2000

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Given a unital ring R and a two-sided ideal I of R, we consider the question of determining when a unit of RaI can be lifted to a unit of R. For the wide class of separative exchange ideals I, we show that the only obstruction to lifting invertibles relies on a K-theoretic condition on I. This allows to extend previously known index theories to this context. Using this we can draw consequences for von Neumann regular rings and C Ãalgebras with real rank zero.

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Section: Preliminary Resultsmentioning

confidence: 99%

Lifting units modulo exchange ideals and C* -algebras with real rank zero

Perera¹

2000

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…The following goal in this paper will be to find the largest exchange ideal of a Leavitt path algebra. We know that it exists by [6,Theorem 3.5] and here we will determine exactly which set of vertices generates it. This paper is organized as follows.…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Largest Ideals in Leavitt Path Algebras

et al. 2020

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We identify largest ideals in Leavitt path algebras: the largest locally left/right artinian (which is the largest semisimple one), the largest locally left/right noetherian without minimal idempotents, the largest exchange, and the largest purely infinite. This last ideal is described as a direct sum of purely infinite simple pieces plus purely infinite non-simple and non-decomposable pieces. The invariance under ring isomorphisms of these ideals is also studied.

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“…For the class of rings generated by idempotents (which is closed by corners), the exchange property is stable by corners. To show this, suppose first that R is a ring generated by the idempotents it contains, and that every corner of R at an idempotent is an exchange ring; then R is an exchange ring by means of [7,Theorem 3.2]. Now, if R is a ring then every corner of R is exchange, as has been proved in the paragraph before.…”

Section: Morita Equivalence For Rings Without a Unit Element Local Rmentioning

confidence: 86%

“…(Exchange example.) The exchange property (see [7] for the definition) is stable by local algebras at elements, as shown in [7,Theorem 1.4].…”

Section: Morita Equivalence For Rings Without a Unit Element Local Rmentioning

confidence: 99%

Morita Equivalence and Morita Invariant Properties: Applications in the Context of Leavitt Path Algebras

Molina¹,

Hernández

2016

Non-Associative and Non-Commutative Algebra and Operator Theory

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In this paper we prove that two idempotent rings are Morita equivalent if every corner of one of them is isomorphic to a corner of a matrix ring of the other one. We establish the converse (which is not true in general) for σ-unital rings having a σ-unit consisting of von Neumann regular elements. The following aim is to show that a property is Morita invariant if it is invariant under taking corners and under taking matrices. The previous results are used to check the Morita invariance of certain ring properties (being locally left/right artinian/noetherian, being categorically left/right artinian, being an I0-ring and being properly purely infinite) and certain graph properties in the context of Leavitt path algebras (Condition (L), Condition (K) and cofinality). A different proof of the fact that a graph with an infinite emitter does not admit any desingularization is also given.

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Local rings of exchange rings

Cited by 22 publications

References 11 publications

Lifting units modulo exchange ideals and C* -algebras with real rank zero

Lifting units modulo exchange ideals and C* -algebras with real rank zero

Largest Ideals in Leavitt Path Algebras

Morita Equivalence and Morita Invariant Properties: Applications in the Context of Leavitt Path Algebras

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