On Bernstein Algebras Satisfying Chain Conditions

Boudi, Nadia; Zitan, Fouad

doi:10.1080/00927870701327690

Cited by 4 publications

(16 citation statements)

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“…Let H l = 0 (l ≥ 2). We may now replace a i by its expression (l times) in (5) to conclude that a j = 0. This implies that N m = 0, and therefore N = F , which completes the proof.…”

Section: Bernstein Algebras and Locally Nilpotent Nonassociative Algementioning

confidence: 99%

“…In our preceding work [5] it has been proved that a Bernstein algebra A that is Jordan or nuclear is necessarily finite-dimensional whenever it is Noetherian or Artinian. In addition, a counter-example was given to show that the hypothesis that A be Jordan or nuclear is essential in this result.…”

Section: Chain Conditions For Subalgebrasmentioning

confidence: 99%

“…(d.c.c.) condition for ideals contained in ker(ω), and hence it is Noetherian (Artinian) in view of [5,Proposition 2.1]. It follows from [5, Proposition 3.1] that the Bernstein-Jordan algebra A/ann U (U) is finite-dimensional.…”

Section: Chain Conditions For Subalgebrasmentioning

confidence: 99%

“…On the other hand, it is well known that one of the most satisfactory developments of the theory of some varieties of algebras, as associative, Jordan, alternative and Lie algebras, is the structure theory of algebras with chain conditions, and there is presently a substantial bibliography in this subject. Concerning Bernstein algebras, a detailed treatment was done in [5] for Bernstein algebras satisfying chain conditions on ideals. Among many other results in that paper, it was especially proved that for a Bernstein algebra which is Jordan or nuclear, each of the Noetherian and Artinian hypotheses implies finite-dimensionality of the algebra and so nilpotency of the barideal.…”

Section: Introductionmentioning

confidence: 99%

“…In this present article we pursue the study of Bernstein algebras satisfying conditions initiated in [5]. After a first section devoted to preliminaries, we show in Section 2 that an arbitrary Bernstein algebra A satisfying the ascending or descending chain condition subalgebras is finite-dimensional.…”

Section: Introductionmentioning

confidence: 99%

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On Bernstein algebras satisfying chain conditions II

Boudi

Zitan

2019

Communications in Algebra

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Following a previous work with Boudi, we continue to investigate Bernstein algebras satisfying chain conditions. First, it is shown that a Bernstein algebra (A, ω) with ascending or descending chain condition on subalgebras is finitedimensional. We also prove that A is Noetherian (Artinian) if and only if its barideal N = ker(ω) is. Next, as a generalization of Jordan and nuclear Bernstein algebras, we study whether a Noetherian (Artinian) Bernstein algebra A with a locally nilpotent barideal N is finite-dimensional. The response is affirmative in the Noetherian case, unlike in the Artinian case. This question is closely related to a result by Zhevlakov on general locally nilpotent nonassociative algebras that are Noetherian, for which we give a new proof. In particular, we derive that a commutative nilalgebra of nilindex 3 which is Noetherian or Artinian is finite-dimensional. Finally, we improve and extend some results of Micali and Ouattara to the Noetherian and Artinian cases.

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“…Let H l = 0 (l ≥ 2). We may now replace a i by its expression (l times) in (5) to conclude that a j = 0. This implies that N m = 0, and therefore N = F , which completes the proof.…”

Section: Bernstein Algebras and Locally Nilpotent Nonassociative Algementioning

confidence: 99%

Section: Chain Conditions For Subalgebrasmentioning

confidence: 99%

Section: Chain Conditions For Subalgebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations