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

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.

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).

scholarly journals The socle of an alternative ring

1970 ◽  
Vol 14 (4) ◽  
pp. 443-463 ◽  
Author(s):  
M Slater
Keyword(s):  
Alternative Ring

scholarly journals Bol loops

1966 ◽  
Vol 123 (2) ◽  
pp. 341-341 ◽  
Author(s):  
D. A. Robinson
Keyword(s):  
Bol Loops

Some Radical Properties of Jordan Matrix Rings

1979 ◽  
Vol 31 (1) ◽  
pp. 189-196
Author(s):  
Michael Rich
Keyword(s):  
Unique Solution ◽  
Identity Element ◽  
Alternative Ring ◽  
Matrix Rings ◽  
Jordan Ring ◽  
Jordan Matrix ◽  
Image Position ◽  
Canonical Involution

Let A be a ring (not necessarily associative) in which 2x = a has a unique solution for each a ∈ A. Then it is known that if A contains an identity element 1 and an involution j : x ↦ x and if Ja is the canonical involution on An determined by where the ai al−l, 1 ≦ i ≦ n are symmetric elements in the nucleus of A then H(An, Ja), the set of symmetric elements of An, for n ≧ 3 is a Jordan ring if and only if either A is associative or n = 3 and A is an alternative ring whose symmetric elements lie in its nucleus [2, p. 127].

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)).

Neurosteroid Analogues. 11. Alternative Ring System Scaffolds:  γ-Aminobutyric Acid Receptor Modulation and Anesthetic Actions of Benz[f]indenes

2006 ◽  
Vol 49 (15) ◽  
pp. 4595-4605 ◽  
Author(s):  
Jamie B. Scaglione ◽  
Brad D. Manion ◽  
Ann Benz ◽  
Amanda Taylor ◽  
Gregory T. DeKoster ◽  
...  
Keyword(s):  
Ring System ◽  
Aminobutyric Acid ◽  
Alternative Ring ◽  
Acid Receptor ◽  
Receptor Modulation

scholarly journals A Road Network for Freight Transport in Flanders: Multi-Actor Multi-Criteria Assessment of Alternative Ring Ways

2013 ◽  
Vol 5 (10) ◽  
pp. 4222-4246 ◽  
Author(s):  
Levi Vermote ◽  
Cathy Macharis ◽  
Koen Putman
Keyword(s):  
Road Network ◽  
Freight Transport ◽  
Alternative Ring

Differential equations of smooth loops related to some space-time models: Integrability conditions and geometry

2018 ◽  
Vol 27 (07) ◽  
pp. 1841004
Author(s):  
L. Sbitneva
Keyword(s):  
Differential Equations ◽  
Homogeneous Spaces ◽  
Space Time ◽  
Lie Triple System ◽  
Lie Group Theory ◽  
Moufang Loops ◽  
Integrability Conditions ◽  
Curvature And Torsion ◽  
The Right ◽  
Bol Loops

The original approach of Lie to the theory of transformation groups acting on smooth manifolds, on the basis of differential equations, being applied to smooth loops, has permitted the development of the infinitesimal theory of smooth loops generalizing the Lie group theory. A loop with the law of associativity verified for its binary operation is a group. It has been shown that the system of differential equations characterizing a smooth loop with the right Bol identity and the integrability conditions lead to the binary-ternary algebra as a proper infinitesimal object, which turns out to be the Bol algebra (i.e. a Lie triple system with an additional bilinear skew-symmetric operation). There exist the analogous considerations for Moufang loops. We will consider the differential equations of smooth loops, generalizing smooth left Bol loops, with the identities that are the characteristic identities for the algebraic description of some relativistic space-time models. Further examinations of the integrability conditions for the differential equations allow us to introduce the proper infinitesimal object for some subclass of loops under consideration. The geometry of corresponding homogeneous spaces can be described in terms of tensors of curvature and torsion.

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