About j{\mathscr{j}}-Noetherian rings

Let R R be a commutative ring with identity and j {\mathscr{j}} an ideal of R R . An ideal I I of R R is said to be a j {\mathscr{j}} -ideal if I ⊈ j I\hspace{0.33em} \nsubseteq \hspace{0.33em}{\mathscr{j}} . We define R R to be a j {\mathscr{j}} -Noetherian ring if each j {\mathscr{j}} -ideal of R R is finitely generated. In this work, we study some properties of j {\mathscr{j}} -Noetherian rings. More precisely, we investigate j {\mathscr{j}} -Noetherian rings via the Cohen-type theorem, the flat extension, decomposable ring, the trivial extension ring, the amalgamated duplication, the polynomial ring extension, and the power series ring extension.