Semi-prime Bernstein algebras (Q1101820)

Let \(A\) be a nonassociative algebra over a field \(K\). \(A\) is called semiprime if it satisfies the following condition: If \(I\) is an ideal of \(A\) and \(I^2=0\), then \(I=0\). \(A\) is called a Bernstein algebra if \(A\) is commutative and has a nontrivial algebra homomorphism \(w: A\to K\) which satisfies the identity \(x^2x^2=w(x)^2x^2\) for all \(x\in A\). The authors prove that: (i) a semiprime Bernstein algebra is a Jordan algebra, (ii) a finitely generated semiprime Bernstein algebra is a field, and (iii) if \(A\) is a finitely generated Bernstein algebra, then the kernel of \(w\) is solvable. The proofs require characteristic different from two.