On weakly coherent rings (Q457935)

In the paper under review, the authors introduce and study the notion of a ``weakly coherent ring''. A commutative ring with identity \(R\) is called a weakly coherent ring if any finitely generated ideal of \(R\) contained in a finitely presented proper ideal of \(R\) is itself finitely presented. All coherent rings are weakly coherent. But using the trivial extension, they give an example of a non-coherent weakly coherent ring.NEWLINENEWLINE The main results indicate that if \(R\) is weakly coherent and \(I\) a finitely generated ideal of \(R\), then \(R/I\) is weakly coherent, and if \((A,M)\) is a quasi local ring and \(E\) an \(A\)-module with \(ME=0\) and let \(R:=A\varpropto E\) be the trivial ring extension of \(A\) by \(E\). Then \(R\) is weakly coherent if and only ifNEWLINENEWLINE (1) \(E\) is an \(A/M\)-vector space with infinite rank; orNEWLINENEWLINE (2) \(E\) is an \(A/M\)-vector space with finite rank and \(A\) is coherent.NEWLINENEWLINE Also they show that a finite direct product of rings is weakly coherent if and only if each of the components is weakly coherent. By an example, they show that the class of weakly coherent rings is not stable under localization.