@nfdc23 I tried to understand your example but Can you provide some other example?

@Neil Strickland But your definition for coherent graded ring in the Def1.2 in your paper is same as I have given for any commutative ring with 1. So I guess this must be true for ungraded one?

@JasonStarr ,@Mohan How do we relate torsion free sheaf and pure sheaf? Pure sheaf implies torsion free sheaf. What is the proof of this statement?

@JasonStarr The definition for pure is as follows: $E $ is pure of dimension d if $dim(F) = d$ for all non-trivial coherent subsheaves $F \subset E$