Four dimensional absolute valued algebras containing a nonzero central idempotent or with left unit

2016 ◽  
Vol 10 ◽  
pp. 513-524
Author(s):  
A. Moutassim ◽  
M. Benslimane
Keyword(s):  
Central Idempotent ◽  
Left Unit

scholarly journals A note on Noetherian orders in Artinian rings

1979 ◽  
Vol 20 (2) ◽  
pp. 125-128 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Noetherian Ring ◽  
Quotient Ring ◽  
Triangular Matrix ◽  
Idempotent Element ◽  
Noetherian Rings ◽  
Central Idempotent ◽  
Matrix Rings ◽  
Regular Elements ◽  
Zero Divisors ◽  
Left And Right

Throughout this note, rings are associative with identity element but are not necessarily commutative. Let R be a left and right Noetherian ring which has an Artinian (classical) quotient ring. It was shown by S. M. Ginn and P. B. Moss [2, Theorem 10] that there is a central idempotent element e of R such that eR is the largest Artinian ideal of R. We shall extend this result, using a different method of proof, to show that the idempotent e is also related to the socle of R/N (where N, throughout, denotes the largest nilpotent ideal of R) and to the intersection of all the principal right (or left) ideals of R generated by regular elements (i.e. by elements which are not zero-divisors). There are many examples of left and right Noetherian rings with Artinian quotient rings, e.g. commutative Noetherian rings in which all the associated primes of zero are minimal together with full or triangular matrix rings over such rings. It was shown by L. W. Small that if R is any left and right Noetherian ring then R has an Artinian quotient ring if and only if the regular elements of R are precisely the elements c of R such that c + N is a regular element of R/N (for further details and examples see [5] and [6]). By the largest Artinian ideal of R we mean the sum of all the Artinian right ideals of R, and it was shown by T. H. Lenagan in [3] that this coincides in any left and right Noetherian ring R with the sum of all the Artinian left ideals of R.

Real division algebras with central idempotent satisfying (x^2, y^2, x^2) = 0

2021 ◽  
Vol 15 (2) ◽  
pp. 69-75
Author(s):  
Bouchra Aharmim ◽  
Kande Diaby ◽  
Oussama Fayz ◽  
Abdellatif Rochdi
Keyword(s):  
Central Idempotent ◽  
Division Algebras

On a Sheaf of Division Rings*

1975 ◽  
Vol 17 (5) ◽  
pp. 727-731
Author(s):  
George Szeto
Keyword(s):  
Principal Ideal ◽  
Regular Ring ◽  
Central Idempotent ◽  
Division Rings ◽  
Maximal Ideals ◽  
Strongly Regular ◽  
Th 1

R. Arens and I. Kaplansky ([1]) call a ring A biregular if every two sided principal ideal of A is generated by a central idempotent and a ring A strongly regular if for any a in A there exists b in A such that a=a2b. In ([1], Sections 2 and 3), a lot of interesting properties of a biregular ring and a strongly regular ring are given. Some more properties can also be found in [3], [5], [8], [9] and [13]. For example, J. Dauns and K. Hofmann ([3]) show that a biregular ring A is isomorphic with the global sections of the sheaf of simple rings A/K where K are maximal ideals of A. The converse is also proved by R. Pierce ([9], Th. 11–1). Moreover, J. Lambek ([5], Th. 1) extends the above representation of a biregular ring to a symmetric module.

scholarly journals Central idempotent measures on connected locally compact groups

1974 ◽  
Vol 15 (1) ◽  
pp. 22-32 ◽  
Author(s):  
Frederick P Greenleaf ◽  
Martin Moskowitz ◽  
Linda Preiss Rothschild
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Central Idempotent ◽  
Locally Compact

scholarly journals Eight-dimensional real absolute-valued algebras with left unit whose automorphism group is trivial

2003 ◽  
Vol 2003 (70) ◽  
pp. 4447-4454 ◽  
Author(s):  
A. Rochdi
Keyword(s):  
Automorphism Group ◽  
Orthogonal Group ◽  
Isomorphism Problem ◽  
Left Unit

We classify, by means of the orthogonal group𝒪7(ℝ), all eight-dimensional real absolute-valued algebras with left unit, and we solve the isomorphism problem. We give an example of those algebras which contain no four-dimensional subalgebras and characterise with the use of the automorphism group those algebras which contain one.

scholarly journals Absolute valued algebras containing a central idempotent

1990 ◽  
Vol 128 (1) ◽  
pp. 180-187 ◽  
Author(s):  
Mohamed Lamei El-Mallah
Keyword(s):  
Central Idempotent

Linear derivations on Banach *-algebras

2021 ◽  
Vol 71 (1) ◽  
pp. 27-32
Author(s):  
Husain Alhazmi ◽  
Abdul Nadim Khan
Keyword(s):  
Positive Integer ◽  
Banach Algebras ◽  
Central Idempotent

Abstract In this paper, it is shown that there is no positive integer n such that the set of x ∈ A $ x\in \mathfrak{A} $ for which [ ( x δ ) n , ( x ∗ δ ) n ( x δ ) n ] ∈ Z ( A ) $ [(x^{\delta})^n, (x^{*{\delta}})^n(x^{\delta})^n]\in \mathcal{Z}(\mathfrak{A}) $ , where δ is a linear derivation on A $ \mathfrak{A} $ or there exists a central idempotent e ∈ Q $ e\in \mathcal{Q} $ such that δ=0 on e Q $ e\mathcal{Q} $ and ( 1 − e ) Q $ (1-e)\mathcal{Q} $ satisfies S 4(X 1, X 2, X 3, X 4). Moreover, we establish other related results.

Some New Class of Absolute Valued Algebras with Left Unit

2010 ◽  
Vol 21 (1) ◽  
pp. 31-40
Author(s):  
M. Benslimane ◽  
A. Moutassim
Keyword(s):  
New Class ◽  
Left Unit
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