A negative answer to a Bahturin-Regev Conjecture about regular algebras in positive characteristic

Let A = A1 ⊕···⊕Ar be a decomposition of the algebra A as a direct sum of vector subspaces. If for every choice of the indices 1 ≤ ij ≤ r there exist aij ∈ Aij such that the product ai1 ···ain = 0, and for every 1 ≤ i,j ≤ r there is a constant β(i,j)= 0 with aiaj = β(i, j)ajai for ai ∈ Ai, aj ∈ Aj, the above decomposition is regular. Bahturin and Regev raised the following conjecture: suppose the regular decomposition comes from a group grading on A, and form the r × r matrix whose (i,j)th entry equals β(i,j). Then this matrix is invertible if and only if the decomposition is minimal (that is one cannot get a regular decomposition of A by coarsening the decomposition). Aljadeff and David proved that the conjecture is true in the case the base field is of characteristic 0. We prove that the conjecture does not hold for algebras over fields of positive characteristic, by constructing algebras with minimal regular decompositions such that the associated matrix is singular.