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 Commutative feebly nil-clean group rings

2019 ◽  
Vol 11 (2) ◽  
pp. 264-270
Author(s):  
Peter V. Danchev
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Group Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Clean Rings ◽  
Unital Ring ◽  
Necessary And Sufficient

Abstract An arbitrary unital ring R is called feebly nil-clean if any its element is of the form q + e − f, where q is a nilpotent and e, f are idempotents with ef = fe. For any commutative ring R and any abelian group G, we find a necessary and sufficient condition when the group ring R(G) is feebly nil-clean only in terms of R, G and their sections. Our result refines establishments due to McGovern et al. in J. Algebra Appl. (2015) on nil-clean rings and Danchev-McGovern in J. Algebra (2015) on weakly nil-clean rings, respectively.

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.

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.

THE NUMBER OF IDEMPOTENTS IN COMMUTATIVE GROUP RINGS

2012 ◽  
Vol 11 (05) ◽  
pp. 1250084
Author(s):  
P. V. DANCHEV
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Commutative Group ◽  
Group Rings ◽  
Arbitrary Characteristic ◽  
Unital Ring

Let R be a commutative unital ring of arbitrary characteristic and let G be a multiplicative Abelian group. For the group ring RG we completely calculate the number (finite or infinite) of its idempotents only in terms of R, G and their sections. This strengthens our previous results in Sarajevo J. Math. (2011) and Filomat (2012).

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 Simplicity of Partial Skew Group Rings of Abelian Groups

2014 ◽  
Vol 57 (3) ◽  
pp. 511-519 ◽  
Author(s):  
Daniel Gonçalves
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Compact Set ◽  
Group Rings ◽  
Partial Action ◽  
Skew Group Rings

AbstractLet A be a ring with local units, E a set of local units for A, G an abelian group, and α a partial action of G by ideals of A that contain local units. We show that A*αG is simple if and only if A is G-simple and the center of the corner eδ0(A*αGe)eδ0 is a field for all e ∊ E. We apply the result to characterize simplicity of partial skew group rings in two cases, namely for partial skew group rings arising from partial actions by clopen subsets of a compact set and partial actions on the set level.

scholarly journals The number of idempotents in abelian group rings

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 719-723
Author(s):  
Peter Danchev
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Group Rings ◽  
Arbitrary Characteristic

Suppose that R is a commutative unitary ring of arbitrary characteristic and G is a multiplicative abelian group. Our main theorem completely determines the cardinality of the set id(RG), consisting of all idempotent elements in the group ring RG. It is explicitly calculated only in terms associated with R, G and their divisions. This result strengthens previous estimates obtained in the literature recently.

Functors on Finite Vector Spaces and Units in Abelian Group Rings

1986 ◽  
Vol 29 (1) ◽  
pp. 79-83 ◽  
Author(s):  
Klaus Hoechsmann
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Elementary Abelian Group ◽  
Vector Spaces ◽  
Group Rings ◽  
Cyclic Subgroups ◽  
Group Of Units ◽  
Finite Vector ◽  
Finite Vector Spaces

AbstractIf A is an elementary abelian group, let (A) denote the group of units, modulo torsion, of the group ring Z[A]. We study (A) by means of the compositewhere C and B run over all cyclic subgroups and factor-groups, respectively. This map, γ, is known to be injective; its index, too, is known. In this paper, we determine the rank of γ tensored (over Z);with various fields. Our main result depends only on the functoriality of

scholarly journals Simple semigroup graded rings

2015 ◽  
Vol 14 (07) ◽  
pp. 1550102 ◽  
Author(s):  
Patrik Nystedt ◽  
Johan Öinert
Keyword(s):  
Simple Semigroup ◽  
Sufficient Conditions ◽  
Group Rings ◽  
Crossed Products ◽  
Graded Rings ◽  
Unital Ring ◽  
Nonzero Idempotent ◽  
Hypercentral Group ◽  
Necessary And Sufficient ◽  
Skew Group Ring

We show that if R is a, not necessarily unital, ring graded by a semigroup G equipped with an idempotent e such that G is cancellative at e, the nonzero elements of eGe form a hypercentral group and Re has a nonzero idempotent f, then R is simple if and only if it is graded simple and the center of the corner subring f ReGe f is a field. This is a generalization of a result of Jespers' on the simplicity of a unital ring graded by a hypercentral group. We apply our result to partial skew group rings and obtain necessary and sufficient conditions for the simplicity of a, not necessarily unital, partial skew group ring by a hypercentral group. Thereby, we generalize a very recent result of Gonçalves'. We also point out how Jespers' result immediately implies a generalization of a simplicity result, recently obtained by Baraviera, Cortes and Soares, for crossed products by twisted partial actions.

Commutative group rings that are présimplifiable or domainlike

2017 ◽  
Vol 16 (01) ◽  
pp. 1750019 ◽  
Author(s):  
D. D. Anderson ◽  
O. A. Al-Mallah
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Commutative Group ◽  
Primary Ideal ◽  
Torsion Subgroup ◽  
Group Rings ◽  
Torsion Free

A commutative ring [Formula: see text] is called présimplifiable (respectively, domainlike) if whenever [Formula: see text] with [Formula: see text], then either [Formula: see text] or [Formula: see text] is a unit in [Formula: see text] (respectively, [Formula: see text] is a primary ideal of [Formula: see text]). Let [Formula: see text] be a commutative ring and [Formula: see text] be a nonzero abelian group. For the group ring [Formula: see text], we prove that if [Formula: see text] is torsion, then [Formula: see text] is présimplifiable (respectively, domainlike) if and only if [Formula: see text] is présimplifiable (respectively, domainlike) and [Formula: see text] is [Formula: see text]-primary with [Formula: see text] (respectively, [Formula: see text]). If [Formula: see text] is torsion-free, then [Formula: see text] is présimplifiable if and only if [Formula: see text] is domainlike if and only if [Formula: see text] is domainlike. Finally, if [Formula: see text] is mixed, [Formula: see text] is présimplifiable (respectively, domainlike) if and only if [Formula: see text] is domainlike and the torsion subgroup of [Formula: see text] is [Formula: see text]-primary with [Formula: see text] (respectively, [Formula: see text]).

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