Goldie theory for Jordan algebras.
The main result of the article is the following Theorem. For a Jordan algebra \(J\) the following conditions are equivalent: (i) \(J\) is a classical order in a nondegenerate Artinian Jordan algebra \(Q\); (ii) \(J\) is a nondegenerate Goldie Jordan algebra; i.e., \(J\) has acc on annihilators and has no infinite direct sum of inner ideals; (iii) \(J\) is nondegenerate, satisfies the acc on the annihilators of its elements, and has finite uniform dimension; (iv) \(J\) is nondegenerate, any nonzero ideal of \(J\) contains a uniform element, and \(J\) has finite uniform dimension; (v) \(J\) is nondegenerate, nonsingular, and has finite uniform dimension. Moreover, \(Q\) is simple iff \(J\) is strongly prime. Thus, Zelmanov's version of Goldie's conditions [\textit{E. Zelmanov}, Sib. Math. J. 28, No. 6, 44--52 (1987; Zbl 0638.17012) and 29, No. 4, 68--74 (1988; Zbl 0663.17016)] still characterizes the quadratic Jordan algebras having an Artinian algebra of quotients which is nondegenerate. The authors also study Jordan versions of the main notions of the associative theory, such as the notion of the uniform ideal, uniform element, singular ideal and uniform dimension. In conclusion, the authors prove the following Theorem. For a Jordan algebra \(J\) the following are equivalent: (i) \(J\) is a classical order in a nondegenerate unital Jordan algebra \(Q\) with finite capacity; (ii) \(J\) is nondegenerate, and an inner ideal of \(J\) is essential iff it contains an injective element. Moreover, \(J\) is prime iff \(Q\) is simple.
scientific article; zbMATH DE number 1749639
scientific article