Let A be a commutative ring with \(1\ne 0\) . An ideal I of A is said to be a non-J-ideal if it is not contained in the Jacobson radical of A. The ring A is called non-J-Noetherian if each non-J-ideal is finitely generated. We show that many of the properties of Noetherain rings are true for non-J-Noetherian rings. Among other results we show that A is a non-J-Noetherian ring if and only if the power series ring \(A[[X_1,\cdots ,X_n]]\) is non-J-Noetherian. Also, we study the transfert of the non-J-Noetherian property via Nagata’s idealization. We investigate the relation between nonnil-Noetherian ring and non-J-Noetherian rings and construct non-trivial (not quasi-local) examples of n-dimensional non-J-Noetherian rings for each \(n\ge 2\) .
