Тип публикации: статья из журнала
Год издания: 2022
Идентификатор DOI: 10.1134/S0037446622010074
Ключевые слова: 512.54:512.55, adjoint group, associated lie ring, automorphism group, local automorphism, nil-triangular subalgebra, nonfinitary generalizations, radical ring
Аннотация: Let $ \Gamma $ be a linearly ordered set (chain), and let$ K $ be an associative commutative ring with a unity.We study the module of all matrices over $ K $ withindices in $ \Gamma $ and the submodule $ NT({\Gamma},K) $ of allmatrices with zeros on and above the main diagonal. All finitarymatrices in $ NT({\Gamma},K) $ form a nil-Показать полностьюring. The automorphisms ofthe adjoint group (in particular, Ado’s and McLain’s groups)were already described for a ring $ K $ with no zero divisors.They depend on the group $ {\mathcal{A}}(\Gamma) $ of all automorphismsand antiautomorphisms of $ \Gamma $.We show that $ NT({\Gamma},K) $ is an algebra with the usual matrixproduct iff either (a) $ \Gamma $ is isometric oranti-isometric to the chain of naturals and$ {\mathcal{A}}(\Gamma)=1 $ or (b) $ \Gamma $ is isometric to the chain of integersand $ {\mathcal{A}}(\Gamma) $ is the infinite dihedral group.Any of these algebras is radical but not a nil-ring.When $ K $ is a domain, we find theautomorphism groups of the ring $ {\mathcal{R}}=NT({\Gamma},K) $of the associated Lie ring $ L({\mathcal{R}}) $ and the adjoint group$ G({\mathcal{R}}) $ (Theorem 3). All three automorphism groupscoincide in case (a). In the main case (b) the group $ \operatorname{Aut}{\mathcal{R}} $ has morecomplicated structure, and the indexof each of the groups $ \operatorname{Aut}L({\mathcal{R}}) $and $ \operatorname{Aut}G({\mathcal{R}}) $ is equal to $ 2 $.As a consequence,we prove that every local automorphism of the algebras $ {\mathcal{R}} $and $ L({\mathcal{R}}) $ is a fixed automorphism modulo $ {\mathcal{R}}
Журнал: Siberian Mathematical Journal
Выпуск журнала: Vol. 63, Is. 1
Номера страниц: 87-96
ISSN журнала: 00374466
Издатель: Pleiades journals