Commutative nilpotent subalgebras with nilpotency index n-1 in the algebra of matrices of order n

The paper establishes the existence of an element with nilpotency index n − 1 in an arbitrary nilpotent commutative subalgebra with nilpotency index n − 1 in the algebra of upper niltriangular matrices Nn(F) over a field F with at least n elements for all n ≥ 5, and also, as a corollary, in the full matrix algebra Mn(F). The result implies an improvement with respect to the base field of known classification theorems due to D. A. Suprunenko, R. I. Tyshkevich, and I. A. Pavlov for algebras of the class considered. © 2017 Springer Science+Business Media New York.