Nil-ideals, J-ideals and their generalizations in commutative rings

Let R be a commutative ring with identity element, J(R) its Jacobson radical and Nil(R) its subset of all nilpotent elements. Recall that an ideal I is said to be an n-ideal (resp. a J-ideal) if whenever a, b∈ R and ab∈ I such that a∉ N(R) (resp. a∉ J(R)), then b∈ I. A quasi J-ideal is an ideal I such that I is a J-ideal. In this paper, we unify the notions of prime ideals, primary ideals, n-ideals and J-ideals in the so-called Q-ideals as follows: given I⊆ Q ideals in R, I is a Q-ideal if whenever xy∈ I and x∉ Q, then y∈ I (equivalently, given ideals A and B of R such that AB⊆ I and A⊈ Q, then B⊆ I). Clearly, if Q= I, respectively Q=I, respectively Q= J(R) , respectively Q= N(R) , then I is a Q-ideal if and only if I is prime, respectively a primary ideal, respectively a J-ideal, respectively an n-ideal. We investigate the properties of these notions in different contexts of commutative rings. Precisely, in trivial ring extensions, amalgamations of rings and pullbacks. Examples illustrating the limits and scopes of our results are provided.