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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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On the structure of ideals of the dual algebra of a coalgebra
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by David E. Radford
Trans. Amer. Math. Soc. 198 (1974), 123-137
DOI: https://doi.org/10.1090/S0002-9947-1974-0346002-4

Abstract:

The weak-* topology is seen to play an important role in the study of various finiteness conditions one may place on a coalgebra $C$ and its dual algebra ${C^ \ast }$. Here we examine the interplay between the topology and the structure of ideals of ${C^ \ast }$. The basic theory has been worked out for the commutative and almost connected cases (see [2]). Our basic tool for reducing to the almost connected case is the classical technique of lifting idempotents. Any orthogonal set of idempotents modulo a closed ideal of $\operatorname {Rad} {C^\ast }$ can be lifted. This technique is particularly effective when $C = {C_1}$. The strongest results we obtain concern ideals of $C_1^ \ast$. Using the properties of idempotents we show that ${C_1} = \Sigma _{x,y} {{C_x}\Lambda {C_y}}$ where ${C_x}$ and ${C_y}$ run over the simple subcoalgebras of $C$. Our first theorem states that a coalgebra $C$ is locally finite and ${C_0}$ is reflexive if and only if every cofinite ideal of ${C^ \ast }$ contains a finitely generated dense ideal. We show in general that a cofinite ideal $I$ which contains a finitely generated dense ideal is not closed. (In fact either equivalent condition of the theorem does not imply $C$ reflexive.) The preceding statement is true if $C = {C_1}$, or more importantly if $I \supset \operatorname {Rad} {C^\ast }$ and ${C^ \ast }/I$ is algebraic. The second theorem characterizes the closure of an ideal with cofinite radical which also contains a finitely generated dense ideal.
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Bibliographic Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 198 (1974), 123-137
  • MSC: Primary 16A24
  • DOI: https://doi.org/10.1090/S0002-9947-1974-0346002-4
  • MathSciNet review: 0346002