# A non-semiprime associative algebra with zero weak radical.

Extracta Mathematicae (1997)

- Volume: 12, Issue: 1, page 53-60
- ISSN: 0213-8743

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topHaily, Abdelfattah. "A non-semiprime associative algebra with zero weak radical.." Extracta Mathematicae 12.1 (1997): 53-60. <http://eudml.org/doc/38511>.

@article{Haily1997,

abstract = {The weak radical, W-Rad(A) of a non-associative algebra A, has been introduced by A. Rodríguez Palacios in [3] in order to generalize the Johnson's uniqueness of norm theorem to general complete normed non-associative algebras (see also [2] for another application of this notion). In [4], he showed that if A is a semiprime non-associative algebra with DCC on ideals, then W-Rad(A) = 0. In the first part of this paper we give an example of a non-semiprime associative algebra A with DCC on ideals and W-Rad(A) = 0. As a consequence we shall see that, in the class of all associative algebras, the subclass S = \{A : W-Rad(A) = 0\} is not a semisimple class relative to a radical in the sense of Amitsur-Kurosh. In the second part of this paper, we shall establish the coincidence between the weak radical and the maximal nilpotent ideal in a finite dimensional Jordan algebra.},

author = {Haily, Abdelfattah},

journal = {Extracta Mathematicae},

keywords = {Algebras asociativas; Algebras de Jordan; Anillos; Espacios normados; Grupo nilpotente; quasi-invertible elements; endomorphisms algebras; multiplication algebras; full subalgebras; weak radical; Jordan algebras},

language = {eng},

number = {1},

pages = {53-60},

title = {A non-semiprime associative algebra with zero weak radical.},

url = {http://eudml.org/doc/38511},

volume = {12},

year = {1997},

}

TY - JOUR

AU - Haily, Abdelfattah

TI - A non-semiprime associative algebra with zero weak radical.

JO - Extracta Mathematicae

PY - 1997

VL - 12

IS - 1

SP - 53

EP - 60

AB - The weak radical, W-Rad(A) of a non-associative algebra A, has been introduced by A. Rodríguez Palacios in [3] in order to generalize the Johnson's uniqueness of norm theorem to general complete normed non-associative algebras (see also [2] for another application of this notion). In [4], he showed that if A is a semiprime non-associative algebra with DCC on ideals, then W-Rad(A) = 0. In the first part of this paper we give an example of a non-semiprime associative algebra A with DCC on ideals and W-Rad(A) = 0. As a consequence we shall see that, in the class of all associative algebras, the subclass S = {A : W-Rad(A) = 0} is not a semisimple class relative to a radical in the sense of Amitsur-Kurosh. In the second part of this paper, we shall establish the coincidence between the weak radical and the maximal nilpotent ideal in a finite dimensional Jordan algebra.

LA - eng

KW - Algebras asociativas; Algebras de Jordan; Anillos; Espacios normados; Grupo nilpotente; quasi-invertible elements; endomorphisms algebras; multiplication algebras; full subalgebras; weak radical; Jordan algebras

UR - http://eudml.org/doc/38511

ER -

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