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

Let RR be a commutative ring with identity and j{\mathscr{j}} an ideal of RR. An ideal II of RR is said to be a j{\mathscr{j}}-ideal if I⊈jI\hspace{0.33em} \nsubseteq \hspace{0.33em}{\mathscr{j}}. We define RR to be a j{\mathscr{j}}-Noetherian ring if each j{\mathscr{j}}-ideal of RR 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.