Characterisation of fuzzy semiprime ideals of the ring of integers \(\mathbb{Z}\) (Q1304269)

In this paper, the authors characterize fuzzy semiprime ideals in the ring of integers \(\mathbb{Z}\). They show that a real fuzzy ideal \(\mu\) of \(\mathbb{Z}\) is semiprime if and only if there exist integers \(n_1, n_2,\dots, n_{k-1}\) such that \(n_{k-1}\mid n_{k-2}\mid n_{k-3},\dots, n_2\mid n_1\) and \(n_2,\dots, n_{k-1}\) are square free, \(n_1\) is square free or 0 according as \(\mu_0= \{x\in \mathbb{Z}\mid \mu(x)= \mu(0)\}\) or \(\mu_0= \{0\}\), and for \(\operatorname {Im}(\mu)= \{t_1, t_2,\dots, t_k\}\) with \(t_1> t_2> \dots> t_k\), \(\mu(x)= t_k\) if \(n_{k-1}\nmid x\); \(\mu(x)= t_{k-1}\) if \(n_{k-1}\mid x\), but \(n_{k-2}\nmid x;\dots\); \(\mu(x)= t_2\) if \(n_2\mid x\), but \(n_1\nmid x\); \(\mu(x)= t_1\) if \(n_1\mid x\).