Resumen de A non-Semiprime Associative Algebra with Zero Weak Radical

Abdelfattah Haily

• 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.