Horizontal Linkage Of Modules

Let R be a commutative Noetherian ring.We give an equivalent characterization for two ideals of R being linked by Gorenstein ideals;and over a Gorenstein local ring R,we give an equivalent characterization for two ideals of R being horizontally linked.Let R be a commutative coherent semiregular ring.We introduce the notion of the hor-izontal linkage of a finitely presented R-module.We prove that the horizontal linkage of a finitely presented R-module is stable.Under the assumption that finitely generated projective R-modules are Hopfian,we prove that for any finitely presented R-module M,M is horizontally linked if and only if it is a stable torsionless module.Further-more,if M is stable and the Gorenstein dimension of M is zero,then M is horizontally linked,?M is a stable module and the Gorenstein dimension of ?M is also zero.It generalizes the main results in[15].