Semi-Simple Artinian Rings of Fixed Points

1975 ◽  
Vol 18 (2) ◽  
pp. 189-190 ◽  
Author(s):  
Miriam Cohen ◽  
Susan Montgomery
Keyword(s):  
Finite Group ◽  
Fixed Points ◽  
Group Of Automorphisms ◽  
Artinian Rings ◽  
A Finite Group

Let G be a finite group of automorphisms of the ring R, and let RG denote the ring of fixed points of G in R; that is, RG={x∊R|Xg = x,∀∊ G}. Let |G| denote the order of G. In this note, we prove the following:Theorem.Assume that R has no nilpotent ideals and no |G|-torsion. Then if RG is semi-simple Artinian, R is semi-simple Artinian.

scholarly journals A note on the fixed subring of an FPF ring

1989 ◽  
Vol 40 (1) ◽  
pp. 109-111 ◽  
Author(s):  
John Clark
Keyword(s):  
Finite Group ◽  
Associative Ring ◽  
Finitely Generated ◽  
Group Of Automorphisms ◽  
Direct Sum ◽  
Epimorphic Image ◽  
A Finite Group

An associative ring R with identity is called a left (right) FPF ring if given any finitely generated faithful left (right) R-module A and any left (right) R-module M then M is the epimorphic image of a direct sum of copies of A. Faith and Page have asked if the subring of elements fixed by a finite group of automorphisms of an FPF ring need also be FPF. Here we present examples showing the answer to be negative in general.

Nilpotent Partition-Inducing Automorphism Groups

1981 ◽  
Vol 33 (2) ◽  
pp. 412-420 ◽  
Author(s):  
Martin R. Pettet
Keyword(s):  
Finite Group ◽  
Automorphism Groups ◽  
Group Of Automorphisms ◽  
A Finite Group ◽  
Identity Elements

If A is a group acting on a set X and x ∈ X, we denote the stabilizer of x in A by CA(x) and let Γ(x) be the set of elements of X fixed by CA(x). We shall say the action of A is partitive if the distinct subsets Γ(x), x ∈ X, partition X. A special example of this phenomenon is the case of a semiregular action (when CA (x) = 1 for every x ∈ X so the induced partition is a trivial one). Our concern here is with the case that A is a group of automorphisms of a finite group G and X = G#, the set of non-identity elements of G. We shall prove that if A is nilpotent, then except in a very restricted situation, partitivity implies semiregularity.

scholarly journals On groups admitting a noncyclic abelian automorphism group

1973 ◽  
Vol 9 (3) ◽  
pp. 363-366 ◽  
Author(s):  
J.N. Ward
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Automorphism Group ◽  
Fixed Points ◽  
Free Operator ◽  
A Finite Group

It is shown that a condition of Kurzwell concerning fixed-points of certain operators on a finite group G is sufficient to ensure that G is soluble. The result generalizes those of Martineau on elementary abelian fixed-point-free operator groups.

A special class of functional equations

2018 ◽  
Vol 68 (2) ◽  
pp. 397-404 ◽  
Author(s):  
Ahmed Charifi ◽  
Radosław Łukasik ◽  
Driss Zeglami
Keyword(s):  
Functional Equation ◽  
Finite Group ◽  
Abelian Group ◽  
Special Class ◽  
Functional Equations ◽  
Additive Functions ◽  
Group Of Automorphisms ◽  
Quadratic Equations ◽  
A Finite Group ◽  
Abelian Monoid

Abstract We obtain in terms of additive and multi-additive functions the solutions f, h: S → H of the functional equation $$\begin{array}{} \displaystyle \sum\limits_{\lambda \in \Phi }f(x+\lambda y+a_{\lambda })=Nf(x)+h(y),\quad x,y\in S, \end{array} $$ where (S, +) is an abelian monoid, Φ is a finite group of automorphisms of S, N = | Φ | designates the number of its elements, {aλ, λ ∈ Φ} are arbitrary elements of S and (H, +) is an abelian group. In addition, some applications are given. This equation provides a joint generalization of many functional equations such as Cauchy’s, Jensen’s, Łukasik’s, quadratic or Φ-quadratic equations.

scholarly journals THE TOM DIECK SPLITTING THEOREM IN EQUIVARIANT MOTIVIC HOMOTOPY THEORY

2021 ◽  
pp. 1-70
Author(s):  
DAVID GEPNER ◽  
JEREMIAH HELLER
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Fixed Points ◽  
Homotopy Theory ◽  
Independent Interest ◽  
Splitting Theorem ◽  
Motivic Homotopy Theory ◽  
Group A ◽  
Motivic Homotopy ◽  
A Finite Group

Abstract We establish, in the setting of equivariant motivic homotopy theory for a finite group, a version of tom Dieck’s splitting theorem for the fixed points of a suspension spectrum. Along the way we establish structural results and constructions for equivariant motivic homotopy theory of independent interest. This includes geometric fixed-point functors and the motivic Adams isomorphism.

scholarly journals Actions of finite groups on commutative rings I

2003 ◽  
Vol 34 (3) ◽  
pp. 201-212
Author(s):  
A. G. Naoum ◽  
W. K. Al-Aubaidy
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Finite Groups ◽  
Commutative Rings ◽  
Group Of Automorphisms ◽  
A Finite Group ◽  
The Ideal

Let $R$ be a commutative ring with 1, and let $G$ be a finite group of automorphisms of $R$. Denote by $R^G$ the fixed subring of $G$, and let $I$ be a subset of $R^G$. In this paper we prove that if the ideal generated by $I$ in $R$ satisfies a certain property with regard to projectivity, flatness, multiplication or related concepts, then the ideal generated by $I$ in $R^G$ also satisfies the same property.

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