Primary decomposition in incomplete Noetherian algebras (Q2051026)

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}