Nonfinitary algebras and their automorphism groups

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 $K$ with indices in $\Gamma$ and the submodule $NT({\Gamma},K)$ of all matrices with zeros on and above the main diagonal. All finitary matrices in $NT({\Gamma},K)$ form a nil-ring. The automorphisms of the 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 automorphisms and antiautomorphisms of $\Gamma$. We show that $NT({\Gamma}, K)$ is an algebra with the usual matrix product iff either (a) $\Gamma$ is isometric or anti-isometric to the chain of naturals and $\mathcal{A} (\Gamma)=1$ or (b) $\Gamma$ is isometric to the chain of integers and $\mathcal{A} (\Gamma)$ is the infinite dihedral group. Any of these algebras is radical but not a nil-ring. When $K$ is a domain, we find the automorphism 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 groups coincide in case {(a)}. In the main case (b) the group $\operatorname{Aut} \mathcal{R}$ has more complicated structure, and the index of 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 $\mathcal{R}^2$.