On non-\(J\)-Noetherian rings (Q6645837)

All rings considered in this paper are commutative with identity. The authors introduce a new class of rings called non-\(J\)-Noetherian rings. Let \(A\) be a ring. 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. The authors prove among others that \(A\) is non-\(J\)-Noetherian if and only if each non-\(J\)-prime ideal of \(A\) is finitely generated. Non-\(J\)-Noetherian rings with divided Jacobson radical are especially investigated. They also study the non-\(J\)-Noetherian property in trivial extensions and give examples of n-dimensional non-\(J\)-Noetherian rings which are neither nonnil-Noetherian nor quasi-local.