Nonfinitary Algebras and Their Automorphism Groups

Abstract

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 ${\mathcal{R}}^{2}$.