Commutative center = center in a prime finitely generated right alternative ring

1997 ◽  
Vol 25 (10) ◽  
pp. 3147-3153 ◽  
Author(s):  
Irvin R. Hentzel ◽  
Erwin Kleinfeld ◽  
Harry F. Smith
Keyword(s):  
Finitely Generated ◽  
Alternative Ring ◽  
Right Alternative Ring

scholarly journals RIGHT NUCLEUS IN GENERALIZED RIGHT ALTERNATIVE RINGS

2015 ◽  
Vol 3 (1) ◽  
pp. 1-12
Author(s):  
Jayalakshmi ◽  
S. Madhavi Latha
Keyword(s):  
Alternative Algebra ◽  
Finitely Generated ◽  
Alternative Ring ◽  
Locally Nilpotent ◽  
The Right ◽  
Right Alternative Ring

Some properties of the right nucleus in generalized right alternative rings have been presented in this paper. In a generalized right alternative ring R which is finitely generated or free of locally nilpotent ideals, the right nucleus Nr equals the center C. Also, if R is prime and Nr ¹ C, then the associator ideal of R is locally nilpotent. Seong Nam [5] studied the properties of the right nucleus in right alternative algebra. He showed that if R is a prime right alternative algebra of char. ≠ 2 and Right nucleus Nr is not equal to the center C, then the associator ideal of R is locally nilpotent. But the problem arises when it come with the study of generalized right alternative ring as the ring dose not absorb the right alternative identity. In this paper we consider our ring to be generalized right alternative ring and try to prove the results of Seong Nam [5]. At the end of this paper we give an example to show that the generalized right alternative ring is not right alternative.

scholarly journals Some properties of the Levitzki radical in alternative rings

1982 ◽  
Vol 32 (1) ◽  
pp. 52-60 ◽  
Author(s):  
Michael Rich
Keyword(s):  
Nilpotent Ideal ◽  
Finitely Generated ◽  
Alternative Ring ◽  
Locally Nilpotent ◽  
Torsion Free

AbstractTwo local nilpotent properties of an associative or alternative ringAcontaining an idempotent are shown. First, ifA=A11+A10+A01+A00is the Peirce decomposition ofArelative toethen ifais associative or semiprime alternative and 3-torsion free then any locally nilpotent idealBofAiigenerates a locally nilpotent ideal 〈B〉 ofA. As a consequenceL(Aii) =Aii∩L(A)for the Levitzki radicalL. Also bounds are given for the index of nilpotency of any finitely generated subring of 〈B〉. Second, ifA(x)denotes a homotope ofAthenL(A)⊆L(A(x))and, in particular, ifA(x)is an isotope ofAthenL(A)=L(A(x)).

Bol loops in right alternative ring

2014 ◽  
Vol 2 (2) ◽  
pp. 18
Author(s):  
Jayalakshmi Karamsi ◽  
Chittem Manjula
Keyword(s):  
Alternative Ring ◽  
Bol Loops ◽  
Right Alternative Ring

On prime right alternative rings with commutators in the left nucleus

1994 ◽  
Vol 50 (2) ◽  
pp. 287-298
Author(s):  
Erwin Kleinfeld ◽  
Harry F. Smith
Keyword(s):  
Associative Ring ◽  
Nonzero Ideal ◽  
Alternative Ring ◽  
Simple Ring ◽  
Characteristic 2 ◽  
Right Alternative Ring

A ring is called s–prime if the 2-sided annihilator of a nonzero ideal must be zero. In particular, any simple ring or prime (—1, 1) ring is s–prime. Also, a nonzero s–prime right alternative ring, with characteristic ≠ 2, cannot be right nilpotent. Let R be a right alternative ring with commutators in the left nucleus. Then R is associative in the following cases: (1) R is prime, with characteristic ≠ 2, and has an idempotent e ≠ 1 such that (e, e, R) = 0. (2) R is an algebra over a commutative-associative ring with 1/6, and R is either s–prime, or R is prime and locally (—1,1).

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

Collectives of Automata in Finitely Generated Groups

2020 ◽  
Vol 108 (5-6) ◽  
pp. 671-678
Author(s):  
D. V. Gusev ◽  
I. A. Ivanov-Pogodaev ◽  
A. Ya. Kanel-Belov
Keyword(s):  
Finitely Generated ◽  
Finitely Generated Groups

scholarly journals TEST IDEALS IN RINGS WITH FINITELY GENERATED ANTI-CANONICAL ALGEBRAS – CORRIGENDUM

2016 ◽  
Vol 17 (4) ◽  
pp. 979-980
Author(s):  
Alberto Chiecchio ◽  
Florian Enescu ◽  
Lance Edward Miller ◽  
Karl Schwede
Keyword(s):  
Finitely Generated

Primitivity in representations of polycyclic groups

1980 ◽  
Vol 88 (1) ◽  
pp. 15-31 ◽  
Author(s):  
D. L. Harper
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Nilpotent Group ◽  
Infinite Dimension ◽  
Finitely Generated ◽  
Polycyclic Groups

In an earlier paper (5) we showed that a finitely generated nilpotent group which is not abelian-by-finite has a primitive irreducible representation of infinite dimension over any non-absolute field. Here we are concerned primarily with the converse question: Suppose that G is a polycyclic-by-finite group with such a representation, then what can be said about G?

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