Yu.V. Bekker, D.V. Levchuk, E.A. Sotnikova. Automorphisms of rings of nonfinitary niltriangular matrices ... P. 7-13

Let $K$ be an associative ring with identity, and let $\Gamma$ be an arbitrary linearly ordered set (briefly, chain). Matrices $\alpha=\|a_{ij}\|$ over $K$ with indices $i$ and $j$ from $\Gamma$ with respect to linear operations always form a $K$-module $M(\Gamma, K)$. The matrix multiplication in $M(\Gamma,K)$ is generally not defined if $\Gamma$ is an infinite chain. The finitary matrices in $M(\Gamma,K)$ form a known ring with matrix multiplication and addition. On the other hand, as proved in 2019, for the chain $\Gamma={\mathbb N}$ of natural numbers, the submodule in $M(\Gamma, K)$ of all (lower) niltriangular matrices with matrix multiplication and addition gives a radical ring $NT(\Gamma,K)$. Its adjoint group is isomorphic to the limit unitriangular group. The automorphisms of the group $UT(\infty,K)$ over a field $K$ of order greater than 2 were studied by R. Slowik. In the present paper, it is proved that any infinite chain $\Gamma$ is isometric or anti-isometric to the chain ${\mathbb N}$ or the chain of all integers if $NT(\Gamma,K)$ with matrix multiplication is a ring. When the ring of coefficients $K$ has no divisors of zero, the main theorem shows that the automorphisms of $NT({\mathbb N},K)$ and of the associated Lie ring, as well as of the adjoint group, are standard.

Keywords: radical ring, Chevalley algebra, niltriangular subalgebra, unitriangular group, nonfinitary generalizations, automorphism

Received July 11, 2020

Revised July 22, 2020

Accepted August 10, 2020

Funding Agency: This work is supported by the Krasnoyarsk Mathematical Center, which is financed by the Ministry of Science and Higher Education of the Russian Federation within the project for the establishment and development of regional centers for mathematical research and education (agreement no. 075-02-2020-1534/1).

Julianna Vladimirovna Bekker, doctoral student, Siberian Federal University, Krasnoyarsk, 660041 Russia, e-mail: angel220@bk.ru

Denis Vladimirovich Levchuk , Cand. Sci. (Phys.-Math.), Siberian Federal University, Krasnoyarsk, 660041 Russia, e-mail: Dlevchuk82@mail.ru

Elena Andreevna Sotnikova, graduate student, Siberian Federal University, Krasnoyarsk, 660041, Russia, e-mail: olgaRV520@yandex.ru

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Cite this article as: Yu.V. Bekker, D.V. Levchuk, E.A. Sotnikova. Automorphisms of rings of nonfinitary niltriangular matrices, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 3, pp. 7–13.