Injective cogenerators for a class of Gabriel topologies (Q1324184)

Let \(R\) be a commutative ring. A linear ring topology \({\mathfrak T}\) on \(R\) is a Gabriel topology if, for every ideal \(J\) of \(R\), \(J' = \{r \in R : (J : r)\) is open\} open implies the openness of \(J\). It is known, that if \({\mathfrak T}\) is a Gabriel topology of \(R\), \(M\) is a maximal ideal, then the localization of \({\mathfrak T}\), \({\mathfrak T}_ M = \{I_ M : I \in {\mathfrak T}\}\) is a Gabriel topology of \(R_ M\). It is shown that if \({\mathfrak T}\) is a Gabriel topology of \(R\), and \(C[M]\) an injective cogenerator for \({\mathfrak T}_ M\), for every maximal ideal \(M\), then \(\prod C[M]\) \((M\) runs over maximal ideals of \(R)\) is an injective cogenerator for \({\mathfrak T}\).