On Jacobson modules (Q2712544)

Let \(R\) be a associative and commutative ring with an identity. Recall that a ring \(R\) is called a Jacobson ring if every prime ideal of \(R\) is the intersection of a family of maximal ideals. In the paper, a Jacobson \(R\)-module \(M\) which is the generalization of the Jacobson ring is defined. Namely, let \(P\) be a prime ideal of \(R\), \(M\) be an \(R\)-module, \(\text{Spec}(R)\) be the set of all prime ideals of \(R\), and \(\text{Supp}(M)=\{P\in \text{Spec}(R): M_P\neq 0\}\) (here \(M_P\) is the localization of \(M\) by \(P\)). The set of all maximal members in \(\text{Supp}(M)\) is denoted \(\text{Max Supp}(M)\). An \(R\)-module \(M\) is called a Jacobson \(R\)-module if every member in \(\text{Supp}(M)\) is the intersection of a family of members in \(\text{Max Supp}(M)\). It is clear that every module over a Jacobson ring is Jacobson. It is shown that if every \(R\)-module is a Jacobson \(R\)-module then \(R\) is a Jacobson ring (proposition 1.2). There exists a Jacobson module over a non-Jacobson ring (example 1.3). Equivalence conditions between Artinian (or Noetherian) modules and Jacobson modules (proposition 1.6) are given.NEWLINENEWLINENEWLINEThere are provided criteria for Jacobson modules (theorem 2.2) and:NEWLINENEWLINENEWLINEFor given two rings \(R\) and \(S\), when \(S\) is an integral extension of \(R\), if an \(R\)-module \(M\) is Jacobson then the \(S\)-module \(M\bigotimes_R S\) is Jacobson (theorem 2.5)NEWLINENEWLINE. On the other hand, if every prime ideal of \(\text{Supp}(M)\) is the contraction of a prime ideal of \(\text{Supp}_S(M\bigotimes_R S)\), then the converse holds.