Structure theorems for certain topological rings

Abstract:

A Hausdorff topological ring B is called centrally linearly compact if the open left ideals form a fundamental system of neighborhoods of zero and B is a strictly linearly compact module over its center. A topological ring A is called locally centrally linearly compact if it contains an open, centrally linearly compact subring. For example, a totally disconnected (locally) compact ring is (locally) centrally linearly compact, and a Hausdorff finite-dimensional algebra with identity over a local field (a complete topological field whose topology is given by a discrete valuation) is locally centrally linearly compact. Let A be a Hausdorff topological ring with identity such that the connected component c of zero is locally compact, A/c is locally centrally linearly compact, and the center C of A is a topological ring having no proper open ideals. A general structure theorem for A is given that yields, in particular, the following consequences: (1) If the additive order of each element of A is infinite or squarefree, then $A = {A_0} \times D$ where ${A_0}$ is a finite-dimensional real algebra and D is the local direct product of a family $({A_\gamma })$ of topological rings relative to open subrings $({B_\gamma })$, where each $({A_\gamma })$ is the cartesian product of finitely many finite-dimensional algebras over local fields. (2) If A has no nonzero nilpotent ideals, each ${A_\gamma }$ is the cartesian product of finitely many full matrix rings over division rings that are finite dimensional over their centers, which are local fields. (3) If the additive order of each element of A is infinite or squarefree and if C contains an invertible, topologically nilpotent element, then A is the cartesian product of finitely many finite-dimensional algebras over R, C, or local fields.