An Existence Results on Positive Solutions for a Remarks on k-Torsionless Modules

Abstract

Let R be a commutative Noetherian ring. The k-torsionless modules are defined in [7] as a generalization of torsionless and reflexive modules, i.e., torsionless modules are 1-torsionless and reflexive modules are 2-torsionless. Some properties of torsionless, reflexive, and k-torsionless modules are investigated in this paper. It is proved that if M is an R-module such that G-dimR(M)<∞, then M is k-torsionless if and only if Mp is k-torsionless for p∊Spec(R) with depth (RP)≤k-1, and dephRp (Mp)≥k for p∊Spec(R) with deph(Rp)≥k. Furthermore, by Auslander-Bridger formula, we prove that M is k-torsionless if and only if Mp is k-torsionless for p∊Spec(R) with depth (RP)≤k-1, and G-dimRp(Mp)≤depth(Rp)-k for p∊Spec(R) with deph(Rp)≥k. Also, it is shown that the class of maximal Cohen-Macaulay modules and the class of k-torsionless modules are equivalent over Gorenstein local ring with dimension k. Finally, we provide the necessary and sufficient conditions which led the tensor product of k-torsionless modules to be k-torsionless.