On associative algebras satisfying the identity x⁵=0

Abstract

We study Kuzmin’s conjecture on the index of nilpotency for the variety N il₅ of associative nil-algebras of degree 5. Due to Vaughan-Lee [11] the problem is reduced to that for k-generator N il₅-superalgebras, where k ≤ 5. We confirm Kuzmin’s conjecture for 2-generator superalgebras proving that they are nilpotent of degree 15.