A non-semiprime associative algebra with zero weak radical.

Haily, 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 -