Localization of injective modules over \(w\)-Noetherian rings (Q2856990)

Let \(R\) be a commutative ring. An ideal \(J\) of a commutative ring \(R\) is called a GV-ideal, denoted by \(J\in GV(R)\), if \(J\) is finitely generated and the natural homomorphism \(\varphi:R\rightarrow\text{Hom}_R(J,R)\) is an isomorphism. An \(R\)-module \(M\) is called a GV-torsion-free module if whenever \(Jx=0\) for some \(J\in GV(R)\) and \(x\in M\), then \(x=0\). Let \(M\) be a GV-torsion-free \(R\)-module. Then \(M\) is said to be a \(w\)-module if \(\text{Ext}^1_R(R/J,M)=0\) for any \(J\in GV(R)\). A \(w\)-module \(M\) is called a \(w\)-Noetherian module if \(M\) has the ascending chain condition on \(w\)-submodules on \(M\), and \(R\) is said to be \(w\)-Noetherian if \(R\) itself is a \(w\)-Noetherian module. In the paper under review some characterizations of injective modules over \(w\)-Noetherian rings are given. It is also shown that each localization of a GV-torsion-free injective module over a \(w\)-Noetherian ring is injective.