scholarly journals SIMPLICITY AND CHAIN CONDITIONS FOR ULTRAGRAPH LEAVITT PATH ALGEBRAS VIA PARTIAL SKEW GROUP RING THEORY

2019 ◽  
Vol 109 (3) ◽  
pp. 299-319 ◽  
Author(s):  
DANIEL GONÇALVES ◽  
DANILO ROYER
Keyword(s):  
Group Ring ◽  
Ring Theory ◽  
Group Rings ◽  
Chain Conditions ◽  
Path Algebras ◽  
Skew Group Rings ◽  
Leavitt Path Algebras ◽  
Skew Group Ring

AbstractWe realize Leavitt ultragraph path algebras as partial skew group rings. Using this realization we characterize artinian ultragraph path algebras and give simplicity criteria for these algebras.

Outer Partial Actions and Partial Skew Group Rings

2015 ◽  
Vol 67 (5) ◽  
pp. 1144-1160 ◽  
Author(s):  
Patrik Nystedt ◽  
Johan Öinert
Keyword(s):  
Abelian Group ◽  
Dynamical Systems ◽  
Group Ring ◽  
Group Rings ◽  
Partial Action ◽  
Unital Ring ◽  
Different Types ◽  
Outer Action ◽  
Skew Group Rings ◽  
Skew Group Ring

AbstractWe extend the classical notion of an outer action α of a group G on a unital ring A to the case when α is a partial action on ideals, all of which have local units. We show that if α is an outer partial action of an abelian group G, then its associated partial skew group ring A *α G is simple if and only if A is G-simple. This result is applied to partial skew group rings associated with two different types of partial dynamical systems.

scholarly journals Leavitt Path Algebras as Partial Skew Group Rings

2014 ◽  
Vol 42 (8) ◽  
pp. 3578-3592 ◽  
Author(s):  
Daniel Gonçalves ◽  
Danilo Royer
Keyword(s):  
Group Rings ◽  
Path Algebras ◽  
Skew Group Rings ◽  
Leavitt Path Algebras

scholarly journals Crossed product Leavitt path algebras

2021 ◽  
pp. 1-21
Author(s):  
Roozbeh Hazrat ◽  
Lia Vaš
Keyword(s):  
Group Ring ◽  
Matrix Ring ◽  
Finite Graph ◽  
Path Algebra ◽  
Crossed Product ◽  
Ordered Group ◽  
Leavitt Path Algebra ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Skew Group Ring

If [Formula: see text] is a directed graph and [Formula: see text] is a field, the Leavitt path algebra [Formula: see text] of [Formula: see text] over [Formula: see text] is naturally graded by the group of integers [Formula: see text] We formulate properties of the graph [Formula: see text] which are equivalent with [Formula: see text] being a crossed product, a skew group ring, or a group ring with respect to this natural grading. We state this main result so that the algebra properties of [Formula: see text] are also characterized in terms of the pre-ordered group properties of the Grothendieck [Formula: see text]-group of [Formula: see text]. If [Formula: see text] has finitely many vertices, we characterize when [Formula: see text] is strongly graded in terms of the properties of [Formula: see text] Our proof also provides an alternative to the known proof of the equivalence [Formula: see text] is strongly graded if and only if [Formula: see text] has no sinks for a finite graph [Formula: see text] We also show that, if unital, the algebra [Formula: see text] is strongly graded and graded unit-regular if and only if [Formula: see text] is a crossed product. In the process of showing the main result, we obtain conditions on a group [Formula: see text] and a [Formula: see text]-graded division ring [Formula: see text] equivalent with the requirements that a [Formula: see text]-graded matrix ring [Formula: see text] over [Formula: see text] is strongly graded, a crossed product, a skew group ring, or a group ring. We characterize these properties also in terms of the action of the group [Formula: see text] on the Grothendieck [Formula: see text]-group [Formula: see text]

scholarly journals SIMPLE REGULAR SKEW GROUP RINGS

2005 ◽  
Vol 04 (02) ◽  
pp. 127-137 ◽  
Author(s):  
KATHI CROW
Keyword(s):  
Group Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Skew Group Rings ◽  
Necessary And Sufficient ◽  
Skew Group Ring

Given a group G acting on a ring R via α:G → Aut (R), one can construct the skew group ring R*αG. Skew group rings have been studied in depth, but necessary and sufficient conditions for the simplicity of a general skew group ring are not known. In this paper, such conditions are given for certain types of skew group rings, with an emphasis on Von Neumann regular skew group rings. Next the results of the first section are used to construct a class of simple skew group rings. In particular, we obtain a more efficient proof of the simplicity of a certain ring constructed by J. Trlifaj.

scholarly journals On Azumaya Galois extensions and skew group rings

1999 ◽  
Vol 22 (1) ◽  
pp. 91-95
Author(s):  
George Szeto
Keyword(s):  
Galois Group ◽  
Group Ring ◽  
Galois Extension ◽  
Group Rings ◽  
Galois Extensions ◽  
Skew Group Rings ◽  
Skew Group Ring

Two characterizations of an Azumaya Galois extension of a ring are given in terms of the Azumaya skew group ring of the Galois group over the extension and a Galois extension of a ring with a special Galois system is determined by the trace of the Galois group.

scholarly journals Some conditions on fixed rings

1981 ◽  
Vol 4 (1) ◽  
pp. 109-117 ◽  
Author(s):  
Edna E. Reiter
Keyword(s):  
Group Ring ◽  
Chain Conditions ◽  
Skew Group Ring

This paper deals with a question about the ascending and descending chain conditions on two-sided ideals. Using the ideas of the skew group ring, certain results for two-sided ideals are proved.

scholarly journals TRACES ON SEMIGROUP RINGS AND LEAVITT PATH ALGEBRAS

2015 ◽  
Vol 58 (1) ◽  
pp. 97-118 ◽  
Author(s):  
ZACHARY MESYAN ◽  
LIA VAŠ
Keyword(s):  
Commutative Rings ◽  
Inverse Semigroups ◽  
Linear Functions ◽  
Group Rings ◽  
Semigroup Rings ◽  
Matrix Rings ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
The Many

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.

TWO-SIDED CHAIN CONDITIONS IN LEAVITT PATH ALGEBRAS OVER ARBITRARY GRAPHS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250044 ◽  
Author(s):  
GENE ABRAMS ◽  
JASON P. BELL ◽  
PINAR COLAK ◽  
KULUMANI M. RANGASWAMY
Keyword(s):  
Directed Graph ◽  
Path Algebra ◽  
Explicit Description ◽  
Leavitt Path Algebra ◽  
Chain Conditions ◽  
Finite Case ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Arbitrary Graphs

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.

scholarly journals CHAIN CONDITIONS ON ÉTALE GROUPOID ALGEBRAS WITH APPLICATIONS TO LEAVITT PATH ALGEBRAS AND INVERSE SEMIGROUP ALGEBRAS

2018 ◽  
Vol 104 (3) ◽  
pp. 403-411 ◽  
Author(s):  
BENJAMIN STEINBERG
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Algebras ◽  
Chain Conditions ◽  
Path Algebras ◽  
Finite Dimensional ◽  
Unit Space ◽  
Leavitt Path Algebras ◽  
Étale Groupoid ◽  
Totally Disconnected

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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