Nil-ideals, \(J\)-ideals and their generalizations in commutative rings (Q6067057)

Let \(R\) be a commutative ring with identity element, \(J (R)\) its Jacobson radical and \(N(R)\) the ideal of all nilpotent elements of \(R\). The purpose of this paper is to unify the notions of prime ideals, primary ideals, \(n\)-ideals (after \textit{U. Tekir} et al. [Filomat 31, No. 10, 2933--2941 (2017; Zbl 1488.13016)]), \(J\)-ideals (after \textit{H. A. Khashan} and \textit{A. B. Bani-Ata} [Int. Electron. J. Algebra 29, 148--164 (2021; Zbl 1467.13005)]) and quasi \(J\)-ideals (after \textit{H. A. Khashan} and \textit{E. Y. Celikel} [``Weakly J-ideals of Commutative Rings'', Preprint, \url{arXiv:2102.07823}]). To accomplish this task, the author introduces the notion of \(Q\)-ideal as follows: given two (proper) ideals \( I\) and \(Q\) of a commutative ring \(R\), \(I\) is said a \(Q\)-ideal if whenever \(ab \in I\) (\(a, b \in R\)) and \(a \notin Q\), then \(b \in I\). Thus, if \(I=Q \) (respectively, \(\sqrt I =Q\);\ \(J (R)=Q\);\ \(N(R)=Q\)), then \( I\) is a \(Q\)-ideal if and only if \(I\) is a prime (respectively, a primary ideal;\ a \(J\)-ideal;\ an \(n\)-ideal). Recall that a quasi \(J\)-ideal is an ideal \(I\) such that \(\sqrt I\) is a \(J\)-ideal. The author investigates the properties of these notions in different contexts of commutative rings, precisely, in trivial ring extensions, amalgamations of rings and pullbacks. Several examples, illustrating the limits and scopes of the main results, are also provided.