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.

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).

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.

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 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.

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]).

Units in Integral Group Rings for Order pq

1995 ◽  
Vol 47 (1) ◽  
pp. 113-131
Author(s):  
Klaus Hoechsmann
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Finite Index ◽  
Finite Abelian Group ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Group Of Units ◽  
Cyclotomic Units

AbstractFor any finite abelian group A, let Ω(A) denote the group of units in the integral group ring which are mapped to cyclotomic units by every character of A. It always contains a subgroup Y(A), of finite index, for which a basis can be systematically exhibited. For A of order pq, where p and q are odd primes, we derive estimates for the index [Ω(A) : Y(A)]. In particular, we obtain conditions for its triviality.

scholarly journals Warfield p-Invariants in Abelian Group Rings of Characteristic p

2013 ◽  
Vol 31 (2) ◽  
pp. 183
Author(s):  
Peter Danchev
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Group Ring ◽  
General Strategy ◽  
Group Rings ◽  
Finitely Generated ◽  
Characteristic P ◽  
Commutative Group Ring ◽  
Normalized Units ◽  
Finite Direct Product

We calculate Warfield p-invariants Wα,p(V (RG)) of the group of normalized units V (RG) in a commutative group ring RG of prime char(RG) = p in each of the following cases: (1) G0/Gp is finite and R is an arbitrary direct product of indecomposable rings; (2) G0/Gp is bounded and R is a finite direct product of fields; (3) id(R) is finite (in particular, R is finitely generated). Moreover, we give a general strategy for the computation of the above Warfield p-invariants under some restrictions on R and G. We also point out an essential incorrectness in a recent paper due to Mollov and Nachev in Commun. Algebra (2011).

scholarly journals Full Identification of Idempotens in Binary Abelian Group Rings

2017 ◽  
Vol 23 (2) ◽  
pp. 67-75
Author(s):  
Kai Lin Ong ◽  
Miin Huey Ang
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Group Ring ◽  
Minimal Distance ◽  
Group Rings ◽  
Cyclic Groups ◽  
Algebraic Properties ◽  
Abelian Group Ring ◽  
Binary Group

Every code in the latest study of group ring codes is a submodule thathas a generator. Study reveals that each of these binary group ring codes can havemultiple generators that have diverse algebraic properties. However, idempotentgenerators get the most attention as codes with an idempotent generator are easierto determine its minimal distance. We have fully identify all idempotents in everybinary cyclic group ring algebraically using basis idempotents. However, the conceptof basis idempotent constrained the exibilities of extending our work into the studyof identication of idempotents in non-cyclic groups. In this paper, we extend theconcept of basis idempotent into idempotent that has a generator, called a generatedidempotent. We show that every idempotent in an abelian group ring is either agenerated idempotent or a nite sum of generated idempotents. Lastly, we show away to identify all idempotents in every binary abelian group ring algebraically by fully obtain the support of each generated idempotent.

Nil-clean group rings

2016 ◽  
Vol 16 (07) ◽  
pp. 1750135 ◽  
Author(s):  
Serap Sahinkaya ◽  
Gaohua Tang ◽  
Yiqiang Zhou
Keyword(s):  
Abelian Group ◽  
Nilpotent Group ◽  
Group Ring ◽  
Nilpotent Element ◽  
General Situation ◽  
Group Rings ◽  
Locally Finite ◽  
Arbitrary Ring ◽  
Clean Ring ◽  
Nil Clean Ring

An element [Formula: see text] of a ring [Formula: see text] is nil-clean, if [Formula: see text], where [Formula: see text] and [Formula: see text] is a nilpotent element, and the ring [Formula: see text] is called nil-clean if each of its elements is nil-clean. In [W. Wm. McGovern, S. Raja and A. Sharp, Commutative nil clean group rings, J. Algebra Appl. 14(6) (2015) 5; Article ID: 1550094], it was proved that, for a commutative ring [Formula: see text] and an abelian group [Formula: see text], the group ring [Formula: see text] is nil-clean, iff [Formula: see text] is nil-clean and [Formula: see text] is a [Formula: see text]-group. Here, we discuss the nil-cleanness of group rings in general situation. We prove that the group ring of a locally finite [Formula: see text]-group over a nil-clean ring is nil-clean, and that the hypercenter of the group [Formula: see text] must be a [Formula: see text]-group if a group ring of [Formula: see text] is nil-clean. Consequently, the group ring of a nilpotent group over an arbitrary ring is nil-clean, iff the ring is a nil-clean ring and the group is a [Formula: see text]-group.

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