Commutative torsion stable rings
All rings are commutative with identity. A ring R is called stable provided \(Hom(X,E)\neq 0,\) for some R-module X and injective R-module E, implies \(Hom(Y,E)\neq 0\) for any essential submodule Y of X. For example, Noetherian rings and perfect rings are stable but examples are given of stable rings which are neither Noetherian nor perfect. It is shown that some properties of Noetherian rings can be extended to stable rings. For example, any finitely generated ideal of a stable ring has the AR property, and thus if the polynomial ring R[x] is stable then R is Noetherian (theorem 3.2). It is proved that if R is a stable ring with finite global and Gabriel dimensions such that every semiprime ideal is a finite intersection of prime ideals then R is Noetherian. On the other hand, if R is stable, coherent and has Gabriel dimension then R is Noetherian. The authors ask whether every semiprime ideal is a finite intersection of prime ideals for any stable ring R. \(\{\) Reviewer's comment: Theorem 3.2 is an immediate consequence of proposition 1.5 because it is easy to show that if (x) has the AR property in the polynomial ring R[x] then R is Noetherian.\(\}\)
scientific article; zbMATH DE number 3863333
scientific article