Ideals in Graph Algebras

Ruiz, Efren; Tomforde, Mark

doi:10.1007/s10468-013-9421-3

Cited by 24 publications

(20 citation statements)

References 14 publications

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“…Because, we will then have two non-zero graded ideals A = X and B = Y with A B and this is not possible by Lemma , we conclude that the cancellation property holds for non-zero ideals in M . Now the graded ideal M possesses local units, being isomorphic to a Leavitt path algebra of a suitable graph (see [11]). From this, it easy to show that the ideals of M are also the ideals of L. Since M contains every other ideal A of L and is graded (so M A = M ∩ A), we conclude that M is also cancellative.…”

Section: Almost Dedekind Domains and Leavitt Path Algebrasmentioning

confidence: 99%

On Prüfer-like properties of Leavitt path algebras

et al. 2019

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Prüfer domains and subclasses of integral domains such as Dedekind domains admit characterizations by means of the properties of their ideal lattices. Interestingly, a Leavitt path algebra L, in spite of being noncommutative and possessing plenty of zero divisors, seems to have its ideal lattices possess the characterizing properties of these special domains.In [8] it was shown that the ideals of L satisfy the distributive law, a property of Prüfer domains and that L is a multiplication ring, a property of Dedekind domains. In this paper, we first show that L satisfies two more characterizing properties of Prüfer domains which are the ideal versions of two theorems in Elementary Number Theory, namely, for positive integers a, b, c, gcd(a, b) · lcm(a, b) = a · b and a · gcd(b, c) = gcd(ab, ac). We also show that L satisfies a characterizing property of almost Dedekind domains in terms of the ideals whose radicals are prime ideals. Finally, we give necessary and sufficient conditions under which L satisfies another important characterizing property of almost Dedekind domains, namely the cancellative property of its non-zero ideals. 0 2010 Mathematics Subject Classification: 16D25; 13F05

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Section: Almost Dedekind Domains and Leavitt Path Algebrasmentioning

confidence: 99%

On Prüfer-like properties of Leavitt path algebras

et al. 2019

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“…In general, we do not know how to decide semiprojectivity of non-unital graph algebras, but our result can be extended to those non-unital graph C * -algebras that happen to be gauge invariant ideals of unital ones. Indeed, following [DJS03] as amended in [RT14], we will for any ideal given by an admissible pair (H, R) consider the map…”

Section: Corners Subquotients and Extensionsmentioning

confidence: 99%

Semiprojectivity and properly infinite projections in graph C⁎-algebras

Eilers

Katsura

2017

Advances in Mathematics

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“…We shall be using the following generalization of the "hedgehog" graph given in Definition 3.8 to arbitrary graphs (see [26]).…”

Section: A Corner-tree Phenomenamentioning

confidence: 99%

Leavitt path algebras with finitely presented irreducible representations

Rangaswamy

2016

Journal of Algebra

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Let E be an arbitrary graph, K be any field and let L = LK (E) be the corresponding Leavitt path algebra. Necessary and sufficient conditions (both graphical and algebraic) are given under which all the irreducible representations of L are finitely presented. In this case, the graph E turns out to be row-finite and the cycles in E form an artinian partial ordered set under a defined relation ≥. When the graph is E is finite, the above graphical conditions were shown in [7] to be equivalent to LK (E) having finite Gelfand-Kirillov dimension. Examples show that this equivalence no longer holds for infinite graphs and a complete description is obtained of Leavitt path algebras over arbitrary graphs having finite Gelfand-

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Ideals in Graph Algebras

Cited by 24 publications

References 14 publications

On Prüfer-like properties of Leavitt path algebras

On Prüfer-like properties of Leavitt path algebras

Semiprojectivity and properly infinite projections in graph C⁎-algebras

Leavitt path algebras with finitely presented irreducible representations

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