On associative algebras satisfying the identity \(x^5 = 0\)

Abstract

We study Kuzmin's conjecture on the index of nilpotency for the variety \(\mathcal{N}  il_5\)  of associative nil-algebras of degree 5. Due to Vaughan-Lee [11] the  problem is reduced to that for \(k\)-generator \(\mathcal{N} il_5\)-superalgebras, where \(k\leq 5\). We confirm Kuzmin's conjecture for 2-generator superalgebras proving that they are nilpotent of degree 15.