Local centrally essential subalgebras of triangular algebras

ABSTRACT

We study local centrally essential subalgebras in the algebra of all upper triangular matrices over a field of characteristic $\ne 2$. It is proved that the algebras of upper triangular $3\times 3$ or $4\times 4$ matrices have only commutative local centrally essential subalgebras. Every algebra of upper triangular matrices of order exceeding 6 contains a non-commutative local centrally essential subalgebra.

KEYWORDS:

• Centrally essential algebra
• algebra of upper triangular matrices
• nilpotence index

MSC2010 DATABASE:

• 16R99
• 20K30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1. Not necessarily unital rings with $(\ast )$ are considered in [13].

2. See, for example, [15, Theorem 6.2.1].

Additional information

Funding

The work of O.V. Lyubimtsev is done under the State assignment No 0729-2020-0055. A.A. Tuganbaev is supported by Russian Scientific Foundation, project 16-11-10013P.