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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Structure theorems for certain topological rings
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by James B. Lucke and Seth Warner PDF
Trans. Amer. Math. Soc. 186 (1973), 65-90 Request permission

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.
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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 186 (1973), 65-90
  • MSC: Primary 16A60
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0325713-X
  • MathSciNet review: 0325713