The canonical topology of certain rings (Q1076776)

Let R be a ring with 1, \(E_ 0\) the injective envelope of the left module \({}_ RR\), \(E_ 1\) the injective envelope of \(E_ 0/R\) and let \(F^ 1_ R\) denote the Gabriel topology corresponding to the hereditary torsion theory cogenerated by \(E_ 0\oplus E_ 1\). The following result giving the description of the topology \(F^ 1_ R\) is proved: Let R be a prime Goldie ring having a family of hereditary Noetherian partial quotient rings \(R_{\lambda}\), \(\lambda\in \Lambda\) such that \(R=\cap_{\lambda \in \Lambda}R_{\lambda}\), then \(F^ 1_ R=\{I:\) I is a left ideal in R, \(I^*=R\}\), where \(I^*=Hom(I,R)\). The terminology see in \textit{B. Stenström}'s book ''Rings of quotients'' (1975; Zbl 0296.16001).