On generalisations of almost prime and weakly prime ideals (Q462970)

Let \(R\) be a commutative ring with nonzero identity. We will denote the set of ideals of \(R\) by \({\mathfrak{I}}(R)\). By a proper ideal \(I\) of \(R\) we mean an ideal \(I\) of \(R\) with \(I\neq R\). Denote by \({\mathfrak{I}}^*(R)\) the set of proper ideals of \(R\). The concept of weakly prime ideals was introduced by \textit{D. D. Anderson} and \textit{E. Smith} [Houston J. Math. 29, No. 4, 831--840 (2003; Zbl 1086.13500)], where an ideal \(P\in {\mathfrak{I}}^*(R)\) is called weakly prime if, for \(a, b\in R\) with \(0\neq ab\in P\), either \(a\in P\) or \(b\in P\). In 2005, \textit{S. M. Bhatwadekar} and \textit{P. K. Sharma} [Commun. Algebra 33, No. 1, 43--49 (2005; Zbl 1072.13003)] defined a proper ideal \(I\) of an integral domain \(R\) to be almost prime (resp., \(n\)-almost prime) if for \(a, b\in R\) with \(ab\in I - I^2\), (resp., \(ab\in I - I^n (n\geq 2\))) either \(a\in I\) or \(b\in I\). This definition can obviously be made for any commutative ring \(R\). Thus: \(I\) prime \(\Rightarrow\) \(I\) weakly prime \(\Rightarrow\) \(I\) \(n\)-almost prime \(\Rightarrow\) \(I\) almost prime. Later, \textit{D. D. Anderson} and \textit{M. Bataineh} [Commun. Algebra 36, No. 2, 686--696 (2008; Zbl 1140.13005)] gave a generalization of prime ideals which covers all the above mentioned definitions. Let \(\phi : {\mathfrak{I}}(R) \to {\mathfrak{I}}(R)\cup \{\emptyset\}\) be a function. A proper ideal \(I\) of \(R\) is said to be \(\phi\)-prime if for \(a, b\in R\) with \(ab\in I - \phi(I)\), either \(a\in I\) or \(b\in I\). In the paper under review, the author gives some other generalizations of prime ideals in commutative rings. A proper ideal \(P\) of \(R\) is said to be an \((n-1, n)\)-\(\Phi_m\)-prime (resp., \((n-1, n)\)-weakly prime) ideal if for \(a_1,\cdots, a_n\in R\), \(a_1\cdots a_n\in P\setminus P^m (resp., a_1\cdots a_n\in P\setminus\{0\}\)) implies that \(a_1\cdots a_{i-1}a_{i+1}\cdots a_n\in P\), for some \(i\in \{1, \cdots, n\}\); (\(m, n\geq 2\)). She studies some basic properties of these classes of ideals.