Automorphisms of rings of nonfinitary niltriangular matrices

J. V. Bekker ; D. V. Levchuk ; E. A. Sotnikova

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 3, pp. 7-13

Résumé

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.

Détail

Keywords: radical ring, Chevalley algebra, unitriangular group, nonfinitary generalizations
Mots-clés : niltriangular subalgebra, automorphism.