A note on Bernstein algebras

Abstract

A commutative algebra A over the field F, endowed with a nonzero homomorphism ω: A→ F, is Bernstein if the identity ( x 2) 2= ω( x) 2 x 2 holds in A. The kernel B of ω is an ideal of codimension 1 satisfying the identity ( x 2) 2=0. In this note we study the inverse problem: Given a commutative algebra over F with the identity ( x 2) 2=0, classify all Bernstein algebras A having B as the kernel of ω. Results are obtained when dim B 2=1. Write B 2= Fc, for some c ϵ B. If xy= b( x, y) c, then b is a symmetric bilinear form on the vector space B. So B can be decomposed as B= rad B> ⊥ B 1, where B 1 is a regular space. A complete solution is given when B 1 is an anisotropic space.