Primitivity in finitely generated special quadratic Jordan algebras
In this interesting paper it is proved that a finitely generated special Jordan algebra \(J\) (over an arbitrary ring of scalars) having a \(*\)- primitive \(*\)-envelope is primitive. The proof of this result is divided into two cases. If \(J\) has nonzero hermitian part then an extension of the Herstein second construction shows that any nonzero ideal of \(J\) contains a nonzero ample ideal. This is the key tool to see that \(J\) is primitive in this case. On the other hand, anti-hermitian finitely generated special Jordan algebras having a \(*\)-primitive \(*\)-envelope are actually finite-dimensional central simple Jordan algebras, and therefore primitive. Since \(*\)-tight envelopes of primitive special Jordan algebras are \(*\)- primitive, the author finally obtains that a finitely generated special Jordan algebra is primitive if and only if any of its \(*\)-tight envelopes is \(*\)-primitive.
scientific article; zbMATH DE number 703023