A structure theorem is given for nondegenerate Jordan algebras J satisfying the ascending chain condition on annihilators of a single element and such that J contains no infinite direct sum of inner deals inside the inner ideal generated by each element x g J. As a consequence of this theorem and of the main results of a Ž Ž . . previous paper by the authors J. Algebra 174 1995 , 1024᎐1048 , it is obtained that such Jordan algebras J are precisely the local orders in nondegenerate Jordan algebras satisfying dcc on principal inner ideals and without non-artinian quadratic ideals, which extends to local orders the Zel'manov theorem for Goldie Jordan algebras. All rights of reproduction in any form reserved. J J Ž . Ž . As a consequence of ii , if J satisfies acc on ann x , x g B, then J satisfies J Ž . acc on ann x , x g B. J Ž . Proof. i Let x g J be such that U s 0. Then x U B ; B l ann B s 0 Ž . x Ž . Ž . Ž . by 8 , which implies x g ann B by 7 . FERNANDEZ LOPEZ AND GARCIA RUŚ´54 Ž . Ž . Ž . ii Clearly, if z g ann x then z g ann x . Conversely, if z g J J Ž . Proof. Let B, C be ideals of J such that B l C ; ann I . If both B J Ž . and C are not contained in ann I , then B l I and C l I are non-zero, J and hence B l C l I / 0 by uniformity of I, which is a contradiction Ž . because I l ann I s 0. Thus one of these ideals B or C is contained in J Ž . ann I , as required. J 3. UNIFORM ELEMENTS
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