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2024 | OriginalPaper | Chapter

4. On Algebraic Algebras Without Divisors of Zero Satisfying \((x^p, x^q, x^r)=0\)

Authors : Mohamed Traoré, Alassane Diouf

Published in: Mathematics of Computer Science, Cybersecurity and Artificial Intelligence

Publisher: Springer Nature Switzerland

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Abstract

Let \(\mathcal {A}\) be an algebraic algebra without divisors of zero of degree \(\neq 8\) with a nonzero idempotent e such that \([e, I(\mathcal {A})]=0\) (resp., e is omnipresent). Then the following assertions are equivalent:

(1)

\(\mathcal {A}\) is quadratic with unit \(e.\)

(2)

\(\mathcal {A}\) is power associative.

(3)

\(\mathcal {A}\) satisfies \((x, x^q, x^r)=0\) and e is a generalized left unit.

(4)

\(\mathcal {A}\) satisfies \((x^p, x^q, x)=0\) and e is a generalized right unit.

(5)

\(\mathcal {A}\) satisfies \((x^p, x^q, x^r)=0\) and e is a generalized unit.

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Title: On Algebraic Algebras Without Divisors of Zero Satisfying
Authors: Mohamed Traoré
Alassane Diouf
Publisher: Springer Nature Switzerland
Book: Mathematics of Computer Science, Cybersecurity and Artificial Intelligence
Print ISBN: 978-3-031-66221-8

Electronic ISBN: 978-3-031-66222-5

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-66222-5_4

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