About \(j\)-Noetherian rings (Q6595263)

The authors introduce and investigate the notion of \(j\)-Noetherian rings which is a generalization of nonnil-Noetherian rings and Noetherian rings. Let \(R\) be a ring and \(j\) an ideal of \(R\). An ideal \(I\) of \(R\) is said to be a \(j\)-ideal if \(I\not\subset j\). They define \(R\) to be a \(j\)-Noetherian ring if each \(j\)-ideal of \(R\) is finitely generated.\N\NThe authors establish the Eakin-Nagata-Formanek theorem, the Cohen-type theorem, and the flat extension of \(j\)-Noetherian rings. After that, they investigate some ring extensions of \(j\)-Noetherian rings. More precisely, the authora study the amalgamated duplication, the trivial ring extension construction, polynomial ring extension, and the power series ring extension of \(j\)-Noetherian rings. Finally they also study some properties of polynomial ring and power series over \(j\)-Noetherian rings.