\(\delta (0)\)-ideals of commutative rings (Q6595645)

Let \(R\) be a commutative ring with nonzero identity. Let \(\mathcal{I}(R)\) denote the set of all ideals of \(R\), and let \(\delta: \mathcal{I}(R)\rightarrow \mathcal{I}(R)\) be a function. In the paper under review, the authors introduced the notion of \(\delta(0)\)-ideals as follows. A proper ideal \(I\) of \(R\) is called a \(\delta(0)\)-ideal, if whenever \(a,b\in R\) with \(ab\in I\) and \(a\notin \delta(0)\), we have \(b\in I\). This notion is an extension of some other types of ideals such as \(n\)-ideals. They investigated the basic properties of \(\delta(0)\)-ideals and also, they gave some examples about \(\delta(0)\)-ideals. Recall that a function \(\delta: \mathcal{I}(R)\rightarrow \mathcal{I}(R)\) is called an ideal expansion, if it has the following properties: \(I\subseteq \delta(I)\) and if \(I\subseteq J\) for some ideals \(I, J\) of \(R\), then \(\delta(I)\subseteq\delta(J)\). Also, a proper ideal \(I\) of \(R\) is said to be a \(J\)-ideal if whenever \(ab\in I\) for \(a,b\in R\), we have either \(a\in J(R)\) or \(b\in I\). It is shown that if \(\delta\) be an expansion function of ideals of \(R\) and \(\delta(0)\) be a maximal ideal, then every \(J\)-ideal is a \(\delta(0)\)-ideal. Also, it is proved that if \(I\) is a maximal \(\delta(0)\)-ideal of \(R\) with \(\delta(I)=I\), then \(I=\delta(0)\). In addition, the authors extended the notion of \(\delta(0)\)-ideals to modules, and they defined the \(\delta(0)\)-modules.