When is a Nakayama closure semiprime?
If \(R\) is a commutative ring, a closure operation \(c\) on \(R\) is a function that associates with each ideal \(I\) an ideal \(c(I)=I^{c}\) containing \(I\) such that the inclusion of ideals is preserved and \(c(c(I))=c(I)\). For a local ring \(R\) with maximal ideal \(\mathfrak{m}\), the author studies the connections between semiprime operations (i.e., operations where \((I^c J^c)^c=(IJ)^c\) for every ideals \(I\) and \(J\)), Nakayama operations (i.e., operations for which \(I \subseteq J \subseteq (I+ \mathfrak{m}J)^c\) implies that \(I^c=J^c\)) and Krull operations (\((0)^c=\bigcap_{n \geq 1}(I^n)^c\) for all ideals \(I\)). When \(R\) is a discrete valuation ring, it is proved that a closure operation is Nakayama if and only if it is semiprime and Krull. Further, the author asks whether a Krull semiprime operation is Nakayama in every one dimensional domain. The paper also contains examples of Nakayama operations that are not semiprime and examples of semiprime operations that are not Nakayama.
scientific article; zbMATH DE number 6370371