Fuzzy prime spectrum of a ring (Q1187451)

Let \(R\) and \(R'\) denote commutative rings. A topology is defined on the set of all fuzzy prime ideals of \(R\) and the resulting space, denoted by \(F\text{Spec}R\), is shown to be \(T_ 0\). A base for the topology of \(F\text{Spec}R\) is also obtained. For \(\alpha\in[0,1)\), the subspace \(A=\{\mu\in F\text{Spec}R| \text{Im} \mu=\{1,\alpha\}\}\) is shown to be compact. In addition, the subspace \(A\) turns out to be \(T_ 1\) iff every element of \(A\) is a fuzzy maximal ideal of \(R\). Moreover, if \(R\) is Boolean, then the subspace \(A\) is shown to be compact, Hausdorff and zero dimensional. For a homomorphism \(f\) from \(R\) onto \(R'\), it is shown that \(F\text{Spec}R'\) is homeomorphic to the subset of \(F\text{Spec}R\) consisting of \(f\)-invariant elements of \(F\text{Spec}R\). It is shown that \(F\text{Spec}R\) is irreducible iff the fuzzy nil radical of \(R\) is a fuzzy prime ideal of \(R\); and \(F\text{Spec}R\) is disconnected iff \(R\) has a nontrivial idempotent element.