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Relevant bibliographies by topics / Leavitt path algebras over arbitrary rings

Academic literature on the topic 'Leavitt path algebras over arbitrary rings'

Author: Grafiati

Published: 10 December 2022

Last updated: 28 January 2023

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Journal articles on the topic "Leavitt path algebras over arbitrary rings"

1

Nordstrom, Hans, and Jennifer A. Firkins Nordstrom. "Leavitt path algebras over arbitrary unital rings and algebras." Journal of Algebra and Its Applications 19, no. 06 (May 31, 2019): 2050107. http://dx.doi.org/10.1142/s0219498820501078.

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We expand the work of Tomforde by further extending the construction of Leavitt path algebras (LPAs) over arbitrary associative, unital rings. We show that many of the results over a commutative ring hold in the more general setting, provide some useful generalizations of prior results, and give a definition for an iterated Leavitt path extension in our context.

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Nystedt, Patrik, and Johan Öinert. "Group gradations on Leavitt path algebras." Journal of Algebra and Its Applications 19, no. 09 (August 20, 2019): 2050165. http://dx.doi.org/10.1142/s0219498820501650.

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Given a directed graph [Formula: see text] and an associative unital ring [Formula: see text] one may define the Leavitt path algebra with coefficients in [Formula: see text], denoted by [Formula: see text]. For an arbitrary group [Formula: see text], [Formula: see text] can be viewed as a [Formula: see text]-graded ring. In this paper, we show that [Formula: see text] is always nearly epsilon-strongly [Formula: see text]-graded. We also show that if [Formula: see text] is finite, then [Formula: see text] is epsilon-strongly [Formula: see text]-graded. We present a new proof of Hazrat’s characterization of strongly [Formula: see text]-graded Leavitt path algebras, when [Formula: see text] is finite. Moreover, if [Formula: see text] is row-finite and has no source, then we show that [Formula: see text] is strongly [Formula: see text]-graded if and only if [Formula: see text] has no sink. We also use a result concerning Frobenius epsilon-strongly [Formula: see text]-graded rings, where [Formula: see text] is finite, to obtain criteria which ensure that [Formula: see text] is Frobenius over its identity component.

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3

Rigby, Simon W., and Thibaud van den Hove. "A classification of ideals in Steinberg and Leavitt path algebras over arbitrary rings." Journal of Algebra 588 (December 2021): 200–249. http://dx.doi.org/10.1016/j.jalgebra.2021.08.021.

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4

STEINBERG, BENJAMIN. "CHAIN CONDITIONS ON ÉTALE GROUPOID ALGEBRAS WITH APPLICATIONS TO LEAVITT PATH ALGEBRAS AND INVERSE SEMIGROUP ALGEBRAS." Journal of the Australian Mathematical Society 104, no. 3 (March 28, 2018): 403–11. http://dx.doi.org/10.1017/s1446788717000374.

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The author has previously associated to each commutative ring with unit$R$and étale groupoid$\mathscr{G}$with locally compact, Hausdorff and totally disconnected unit space an$R$-algebra$R\,\mathscr{G}$. In this paper we characterize when$R\,\mathscr{G}$is Noetherian and when it is Artinian. As corollaries, we extend the characterization of Abrams, Aranda Pino and Siles Molina of finite-dimensional and of Noetherian Leavitt path algebras over a field to arbitrary commutative coefficient rings and we recover the characterization of Okniński of Noetherian inverse semigroup algebras and of Zelmanov of Artinian inverse semigroup algebras.

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5

Özdin, Tufan. "On endomorphism rings of Leavitt path algebras." Filomat 32, no. 4 (2018): 1175–81. http://dx.doi.org/10.2298/fil1804175o.

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Let E be an arbitrary graph, K be any field and A be the endomorphism ring of L := LK(E) considered as a right L-module. Among the other results, we prove that: (1) if A is a von Neumann regular ring, then A is dependent if and only if for any two paths in L satisfying some conditions are initial of each other, (2) if A is dependent then LK(E) is morphic, (3) L is morphic and von Neumann regular if and only if L is semisimple and every homogeneous component is artinian.

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6

ABRAMS, GENE, JASON P. BELL, PINAR COLAK, and KULUMANI M. RANGASWAMY. "TWO-SIDED CHAIN CONDITIONS IN LEAVITT PATH ALGEBRAS OVER ARBITRARY GRAPHS." Journal of Algebra and Its Applications 11, no. 03 (May 24, 2012): 1250044. http://dx.doi.org/10.1142/s0219498811005713.

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Let E be any directed graph, and K be any field. For any ideal I of the Leavitt path algebra LK(E) we provide an explicit description of a set of generators for I. This description allows us to classify the two-sided noetherian Leavitt path algebras over arbitrary graphs. This extends similar results previously known only in the row-finite case. We provide a number of additional consequences of this description, including an identification of those Leavitt path algebras for which all two-sided ideals are graded. Finally, we classify the two-sided artinian Leavitt path algebras over arbitrary graphs.

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7

VAŠ, LIA. "GRADED CHAIN CONDITIONS AND LEAVITT PATH ALGEBRAS OF NO-EXIT GRAPHS." Journal of the Australian Mathematical Society 105, no. 2 (December 12, 2017): 229–56. http://dx.doi.org/10.1017/s1446788717000295.

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We obtain a complete structural characterization of Cohn–Leavitt algebras over no-exit objects as graded involutive algebras. Corollaries of this result include graph-theoretic conditions characterizing when a Leavitt path algebra is a directed union of (graded) matricial algebras over the underlying field and over the algebra of Laurent polynomials and when the monoid of isomorphism classes of finitely generated projective modules is atomic and cancelative. We introduce the nonunital generalizations of graded analogs of noetherian and artinian rings, graded locally noetherian and graded locally artinian rings, and characterize graded locally noetherian and graded locally artinian Leavitt path algebras without any restriction on the cardinality of the graph. As a consequence, we relax the assumptions of the Abrams–Aranda–Perera–Siles characterization of locally noetherian and locally artinian Leavitt path algebras.

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8

MESYAN, ZACHARY, and LIA VAŠ. "TRACES ON SEMIGROUP RINGS AND LEAVITT PATH ALGEBRAS." Glasgow Mathematical Journal 58, no. 1 (July 21, 2015): 97–118. http://dx.doi.org/10.1017/s0017089515000087.

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AbstractThe trace on matrix rings, along with the augmentation map and Kaplansky trace on group rings, are some of the many examples of linear functions on algebras that vanish on all commutators. We generalize and unify these examples by studying traces on (contracted) semigroup rings over commutative rings. We show that every such ring admits a minimal trace (i.e., one that vanishes only on sums of commutators), classify all minimal traces on these rings, and give applications to various classes of semigroup rings and quotients thereof. We then study traces on Leavitt path algebras (which are quotients of contracted semigroup rings), where we describe all linear traces in terms of central maps on graph inverse semigroups and, under mild assumptions, those Leavitt path algebras that admit faithful traces.

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9

Bell, Jason P., T. H. Lenagan, and Kulumani M. Rangaswamy. "Leavitt path algebras satisfying a polynomial identity." Journal of Algebra and Its Applications 15, no. 05 (March 30, 2016): 1650084. http://dx.doi.org/10.1142/s0219498816500845.

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Leavitt path algebras [Formula: see text] of an arbitrary graph [Formula: see text] over a field [Formula: see text] satisfying a polynomial identity are completely characterized both in graph-theoretic and algebraic terms. When [Formula: see text] is a finite graph, [Formula: see text] satisfying a polynomial identity is shown to be equivalent to the Gelfand–Kirillov dimension of [Formula: see text] being at most one, though this is no longer true for infinite graphs. It is shown that, for an arbitrary graph [Formula: see text], the Leavitt path algebra [Formula: see text] has Gelfand–Kirillov dimension zero if and only if [Formula: see text] has no cycles. Likewise, [Formula: see text] has Gelfand–Kirillov dimension one if and only if [Formula: see text] contains at least one cycle, but no cycle in [Formula: see text] has an exit.

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10

Pino, G. Aranda, K. M. Rangaswamy, and M. Siles Molina. "Generalized Regularity Conditions for Leavitt Path Algebras over Arbitrary Graphs." Communications in Algebra 42, no. 1 (October 18, 2013): 325–31. http://dx.doi.org/10.1080/00927872.2012.714026.

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Book chapters on the topic "Leavitt path algebras over arbitrary rings"

1

Kanwar, Pramod, Meenu Khatkar, and R. K. Sharma. "Basic One-Sided Ideals of Leavitt Path Algebras over Commutative Rings." In Springer Proceedings in Mathematics & Statistics, 155–65. Singapore: Springer Nature Singapore, 2022. http://dx.doi.org/10.1007/978-981-19-3898-6_12.

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