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

Let be a commutative ring with identity and an ideal of . An ideal of is said to be a -ideal if . We define to be a -Noetherian ring if each -ideal of is finitely generated. In this work, we study some properties of -Noetherian rings. More precisely, we investigate -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.