Some ∗-Clean Group Rings

2015 ◽  
Vol 22 (01) ◽  
pp. 169-180 ◽  
Author(s):  
Yanyan Gao ◽  
Jianlong Chen ◽  
Yuanlin Li
Keyword(s):  
Local Ring ◽  
Group Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Clean Rings ◽  
Clean Ring ◽  
Commutative Local Ring ◽  
Necessary And Sufficient

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection. It is obvious that ∗-clean rings are clean. Vaš asked whether there exists a clean ring with involution that is not ∗-clean. In this paper, we investigate when a group ring RG is ∗-clean, where ∗ is the classical involution on RG. We obtain necessary and sufficient conditions for RG to be ∗-clean, where R is a commutative local ring and G is one of the groups C3, C4, S3 and Q8. As a consequence, we provide many examples of group rings which are clean but not ∗-clean.

On *-clean group rings

2014 ◽  
Vol 14 (01) ◽  
pp. 1550004 ◽  
Author(s):  
Yuanlin Li ◽  
M. M. Parmenter ◽  
Pingzhi Yuan
Keyword(s):  
Local Ring ◽  
Prime Order ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Group Rings ◽  
Clean Ring ◽  
Commutative Local Ring ◽  
Irreducible Factorization ◽  
Necessary And Sufficient

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection. Clearly a *-clean ring is clean. Vaš asked whether there exists a clean ring with involution * that is not *-clean. In a recent paper, Gao, Chen and the first author investigated when a group ring RG with classical involution * is *-clean and obtained necessary and sufficient conditions for RG to be *-clean, where R is a commutative local ring and G is one of C3, C4, S3 and Q8. As a consequence, the authors provided many examples of group rings which are clean, but not *-clean. In this paper, we continue this investigation and we give a complete characterization of when the group algebra 𝔽Cp is *-clean, where 𝔽 is a field and Cp is the cyclic group of prime order p. Our main result is related closely to the irreducible factorization of a pth cyclotomic polynomial over the field 𝔽. Among other results we also obtain a complete characterization of when RCn (3 ≤ n ≤ 6) is *-clean where R is a commutative local ring.

Skew-symmetric Elements in Nonlinear Involutions in Group Rings

2015 ◽  
Vol 22 (02) ◽  
pp. 321-332
Author(s):  
A. P. Raposo
Keyword(s):  
Group Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Necessary And Sufficient ◽  
Nonlinear Extensions

Given an involution in a group G, it can be extended in various ways to an involution in the group ring RG, where R is a ring, not necessarily commutative. In this paper nonlinear extensions are considered and necessary and sufficient conditions are given on the group G, its involution, the ring R and the extension for the set of skew-symmetric elements to be commutative and for it to be anticommutative.

Symmetric Elements of Nonlinear Involutions in Group Rings

2012 ◽  
Vol 19 (spec01) ◽  
pp. 1041-1050 ◽  
Author(s):  
R. García-Delgado ◽  
A. P. Raposo
Keyword(s):  
Group Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Linear Extensions ◽  
Necessary And Sufficient

Given an involution φ : G → G in a group G and a ring R, we study the extensions, not necessarily linear, to an involution ψ : RG → RG in the group ring RG. We investigate the symmetric elements, those α ∈ RG for which ψ(α) = α, and give necessary and sufficient conditions for the set of symmetric elements, (RG)ψ, to be a subring of RG. This work is a generalization of [6] and references therein where only linear extensions of the group involution are considered.

scholarly journals COMMUTATIVITY OF SKEW SYMMETRIC ELEMENTS IN GROUP RINGS

2007 ◽  
Vol 50 (1) ◽  
pp. 37-47 ◽  
Author(s):  
Osnel Broche Cristo ◽  
César Polcino Milies
Keyword(s):  
Commutative Ring ◽  
Group Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Necessary And Sufficient

AbstractLet $R$ be a commutative ring with unity and let $G$ be a group. The group ring $RG$ has a natural involution that maps each element $g\in G$ to its inverse. We denote by $RG^-$ the set of skew symmetric elements under this involution. We study necessary and sufficient conditions for $RG^-$ to be commutative.

scholarly journals Unique factorisation in P.I. group-rings

1995 ◽  
Vol 59 (2) ◽  
pp. 232-243 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Group Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Necessary And Sufficient

AbstractWe shall give necessary and sufficient conditions on the ring R and the group G for the group-ring RG to be a prime P. I. ring with the unique factorisation property as defined in [5].

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.

Group Rings over Z(p) with FC Unit Groups

1980 ◽  
Vol 32 (5) ◽  
pp. 1266-1269 ◽  
Author(s):  
H. Merklen ◽  
C. Polcino Milies
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Group Ring ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Conjugacy Classes ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Necessary And Sufficient

Let RG denote the group ring of a group G over a commutative ring R with unity. We recall that a group is said to be an FC-group if all its conjugacy classes are finite.In [6], S. K. Sehgal and H. Zassenhaus gave necessary and sufficient conditions for U(RG) to be an FC-group when R is either Z, the ring of rational integers, or a field of characteristic 0.One of the authors considered this problem for group rings over infinite fields of characteristic p ≠ 2 in [5] and G. Cliffs and S. K. Sehgal [1] completed the study for arbitrary fields. Also, group rings of finite groups over commutative rings containing Z(p), a localization of Z over a prime ideal (p) were studied in [4].

scholarly journals Normal and subnormal subgroups in the group of units of group rings

1985 ◽  
Vol 31 (3) ◽  
pp. 355-363 ◽  
Author(s):  
Jairo Zacarias Goncalves
Keyword(s):  
Group Ring ◽  
Sufficient Conditions ◽  
Finite Subgroup ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Infinite Field ◽  
Group Of Units ◽  
Necessary And Sufficient ◽  
Subnormal Subgroups

Let KG be the group ring of the group G over the infinite field K, and let U(KG) be its group of units. If G is torsion, we obtain necessary and sufficient conditions for a finite subgroup H of G to be either normal or subnormal in U(KG). Actually, if H is subnormal in U(KG), we can handle not only the case H finite, but the precise assumptions depend on the characteristic of K.

scholarly journals VANISHING OF (CO)HOMOLOGY OVER DEFORMATIONS OF COHEN-MACAULAY LOCAL RINGS OF MINIMAL MULTIPLICITY

2018 ◽  
Vol 61 (03) ◽  
pp. 705-725
Author(s):  
DIPANKAR GHOSH ◽  
TONY J. PUTHENPURAKAL
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Sufficient Conditions ◽  
Regular Sequence ◽  
Necessary And Sufficient Conditions ◽  
Local Rings ◽  
Syzygy Modules ◽  
Tor Modules ◽  
Necessary And Sufficient ◽  
Minimal Multiplicity

AbstractLet R be a d-dimensional Cohen–Macaulay (CM) local ring of minimal multiplicity. Set S := R/(f), where f := f1,. . .,fc is an R-regular sequence. Suppose M and N are maximal CM S-modules. It is shown that if ExtSi(M, N) = 0 for some (d + c + 1) consecutive values of i ⩾ 2, then ExtSi(M, N) = 0 for all i ⩾ 1. Moreover, if this holds true, then either projdimR(M) or injdimR(N) is finite. In addition, a counterpart of this result for Tor-modules is provided. Furthermore, we give a number of necessary and sufficient conditions for a CM local ring of minimal multiplicity to be regular or Gorenstein. These conditions are based on vanishing of certain Exts or Tors involving homomorphic images of syzygy modules of the residue field.

Some Results on Semi-Perfect Group Rings

1974 ◽  
Vol 26 (1) ◽  
pp. 121-129 ◽  
Author(s):  
S. M. Woods
Keyword(s):  
Group Ring ◽  
Polynomial Ring ◽  
Strong Interaction ◽  
Sufficient Conditions ◽  
Group Rings ◽  
Commutative Case ◽  
Perfect Group ◽  
Case Examples ◽  
Complete Answer ◽  
Necessary And Sufficient

The aim of this paper is to find necessary and sufficient conditions on a group G and a ring A for the group ring AG to be semi-perfect. A complete answer is given in the commutative case, in terms of the polynomial ring A[X] (Theorem 5.8). In the general case examples are given which indicate a very strong interaction between the properties of A and those of G. Partial answers to the question are given in Theorem 3.2, Proposition 4.2 and Corollary 4.3.

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