On \(\phi-n\)-absorbing primary ideals of commutative rings (Q2806537)

Let \(R\) be a commutative ring and \(n\) is a positive integer. Let \(\phi: \mathfrak{J}(R)\to \mathfrak{J}(R)\) be a function where \(\mathfrak{J}(R)\) denotes the set of all ideals of \(R\). In the paper under review the authors introduced and studied the concept of \(\phi\)-\(n\)-absorbing primary ideal. An ideal \(I\) of \(R\) is called \textit{\(\phi\)-\(n\)-absorbing primary} if whenever \(a_1,a_2,\ldots,a_{n+1}\in R\) and \(a_1a_2\cdots a_{n+1}\in I\setminus\phi(I)\), either \(a_1a_2\cdots a_{n}\in I\) or the product of \(a_{n+1}\) with \((n-1)\) of \(a_1,a_2,\ldots, a_{n}\) is in \(\sqrt{I}\). This is the generalization of the notion of \(n\)-absorbing ideal introduced in [\textit{D. F. Anderson} and \textit{A. Badawi}, Commun. Algebra 39, No. 5, 1646--1672 (2011; Zbl 1232.13001)]. After proving some basic properties of \(\phi\)-\(n\)-absorbing primary ideals they show that a Noetherian domain \(R\) is a Dedekind domain if and only if a nonzero \(n\)-absorbing primary ideal of \(R\) is in the form of \(I = M^{t_1}_1M^{t_2}_2\cdots M^{t_i}_i\) for some \(1\leq i\leq n\) and some distinct maximal ideals \(M_1,M_2,\ldots,M_i\) of \(R\) and some positive integers \(t_1,t_2,\ldots,t_i\). Also they investigate \(\phi\)-\(n\)-absorbing primary ideals of direct products of commutative rings and with respect to idealization.