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\begin{document}$ \Gamma $\end{document} be a linearly ordered set (chain), and let
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\begin{document}$ K $\end{document} be an associative commutative ring with a unity.
We study the module of all matrices over \documentclass[12pt]{minimal}
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\begin{document}$ K $\end{document} with
indices in \documentclass[12pt]{minimal}
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\begin{document}$ \Gamma $\end{document} and the submodule \documentclass[12pt]{minimal}
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\begin{document}$ NT({\Gamma},K) $\end{document} of all
matrices with zeros on and above the main diagonal. All finitary
matrices in \documentclass[12pt]{minimal}
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\begin{document}$ NT({\Gamma},K) $\end{document} form a nil-ring. The automorphisms of
the adjoint group (in particular, Ado’s and McLain’s groups)
were already described for a ring \documentclass[12pt]{minimal}
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\begin{document}$ K $\end{document} with no zero divisors.
They depend on the group \documentclass[12pt]{minimal}
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\begin{document}$ {\mathcal{A}}(\Gamma) $\end{document} of all automorphisms
and antiautomorphisms of \documentclass[12pt]{minimal}
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\begin{document}$ \Gamma $\end{document}.
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\begin{document}$ NT({\Gamma},K) $\end{document} is an algebra with the usual matrix
product iff either (a) \documentclass[12pt]{minimal}
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\begin{document}$ \Gamma $\end{document} is isometric or
anti-isometric to the chain of naturals and
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\begin{document}$ {\mathcal{A}}(\Gamma)=1 $\end{document} or (b) \documentclass[12pt]{minimal}
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\begin{document}$ \Gamma $\end{document} is isometric to the chain of integers
and \documentclass[12pt]{minimal}
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\begin{document}$ {\mathcal{A}}(\Gamma) $\end{document} is the infinite dihedral group.
Any of these algebras is radical but not a nil-ring.
When \documentclass[12pt]{minimal}
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\begin{document}$ K $\end{document} is a domain, we find the
automorphism groups of the ring \documentclass[12pt]{minimal}
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\begin{document}$ {\mathcal{R}}=NT({\Gamma},K) $\end{document}
of the associated Lie ring \documentclass[12pt]{minimal}
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\begin{document}$ L({\mathcal{R}}) $\end{document} and the adjoint group
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\begin{document}$ G({\mathcal{R}}) $\end{document} (Theorem 3). All three automorphism groups
coincide in case (a). In the main case (b) the group \documentclass[12pt]{minimal}
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\begin{document}$ \operatorname{Aut}{\mathcal{R}} $\end{document} has more
complicated structure, and the index
of each of the groups \documentclass[12pt]{minimal}
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\begin{document}$ \operatorname{Aut}L({\mathcal{R}}) $\end{document}
and \documentclass[12pt]{minimal}
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\begin{document}$ \operatorname{Aut}G({\mathcal{R}}) $\end{document} is equal to \documentclass[12pt]{minimal}
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\begin{document}$ 2 $\end{document}.
As a consequence,
we prove that every local automorphism of the algebras \documentclass[12pt]{minimal}
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\begin{document}$ {\mathcal{R}} $\end{document}
and \documentclass[12pt]{minimal}
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\begin{document}$ L({\mathcal{R}}) $\end{document} is a fixed automorphism modulo \documentclass[12pt]{minimal}
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\begin{document}$ {\mathcal{R}}^{2} $\end{document}.
