Primary decomposition in incomplete Noetherian algebras
type: Item
instance of: scholarly article
title: Primary decomposition in incomplete Noetherian algebras
zbMATH Open document ID: 1481.46044
author: Sara El Kinani
published in: Bollettino dell'Unione Matematica Italiana
publication date: 2021-09-01T00:00:00Z
review text: The paper deals with unital commutative algebras only. A Noetherian algebra is an algebra in which the family of all its ideals satisfies the ascending chain condition. An ideal \(I\) of an algebra \(A\) is primary if from \(x, y\in A\) and \(xy\in I\) it follows that either \(x\in I\) or there exists \(n\in{\mathbb Z}^+\) such that \(y^n\in I\). Let \(I\) be an ideal of a untal commutative algebra \(A\). The radical of \(I\) is the set \[\sqrt{I}=\{x\in A: \exists n\in{\mathbb Z}^+: x^n\in I\}.\] Let \(r\in{\mathbb Z}^+\) and \(Q_1, \dots, Q_r\) be primary ideals of \(A\) such that their radicals \(\sqrt{Q_1}, \dots, \sqrt{Q_r}\) are pairwise different and \(Q_i\) does not contain \(\bigcap_{j\not=i} Q_j\). Then the intersection \(Q_1\cap \dots\cap Q_r\) is called the reduced decomposition. A topological algebra is considered to be a Hausdorff space which is also an algebra with separately continuous multiplication. The main results of the paper are the following: \begin{itemize} \item[(1)] Let \(A\) be a Noetherian commutative unital topological algebra, $I$ a proper closed ideal of \(A\) and \(I=Q_1\cap\dots\cap Q_r\) a reduced decomposition. Then all the radicals \(\sqrt{Q_1}, \dots, \sqrt{Q_r}\) are closed. \item[(2)] Let \((A, \tau)\) be a Noetherian commutative unital topological algebra with all ideals closed and with every maximal ideal having codimension \(1\). Let \((B, \tau')\) be a commutative unital topological algebra with all ideals closed. Then the graph of any morphism \(f:(B, \tau')\rightarrow (A, \tau)\) of algebras is closed. \end{itemize} As an application, the author obtains the result that \begin{itemize} \item[(3)] In a Noetherian commutative unital topological algebra with every maximal ideal of codimension \(1\), there exists at most one topology making this algebra an \(F\)-algebra (i.e., a complete metrizable algebra). \end{itemize}
Mathematics Subject Classification ID: 46J20
Mathematics Subject Classification ID: 13E05
zbMATH DE Number: 7389311
zbMATH Keywords: Noetherian algebra
zbMATH Keywords: Fréchet algebra
zbMATH Keywords: closed graph
zbMATH Keywords: prime ideal
zbMATH Keywords: primary ideal
zbMATH Keywords: topological algebra
zbMATH Keywords: algebra morphism
zbMATH Keywords: primary decomposition
reviewed by: Mart Abel
MaRDI profile type: Publication
full work available at URL: s40574-020-00274-1
OpenAlex ID: W3126404027
cites work: Elements of Finite Closed Descent in a Banach Algebra
cites work: Algèbres topologiques à idéaux à gauche fermés
cites work: Q4180796
cites work: Q4148077
cites work: Closed Ideals in C(X)
cites work: Noetherian Banach algebras are finite dimensional
cites work: A characterization of commutative Fréchet algebras with all ideals closed
DOI: 10.1007/S40574-020-00274-1
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description: scientific article; zbMATH DE number 7389311