On the relation between two definitions of torsion functors

Let R be a commutative ring, and let aR be an ideal. For an R-module we consider the sub-R-modules Γa(M)={xMnN:an(0:Rx)} and ˜Γa(M)={xMa(0:Rx)} of M. These definitions give rise to left exact subfunctors Γa and ˜Γa of the identity functor on the category of R-modules, and Γa is a subfunctor of ˜Γa.

Recall that a subfunctor F of the identity functor is called a radical if F(M/F(M))=0 for every R-module M. Moreover, for every subfunctor F of the identity functor there exists the smallest radical containing F.

Now, one can show that ˜Γa is a radical, while Γa need not be so. I conjecture but am unable to prove the following:

Conjecture: ˜Γa is the smallest radical containing Γa.

Is anything known about this problem?

(Motivation: In the literature about torsion functors and their right derived functors (i.e., local cohomology) both definitions are used. Since most authors work over noetherian rings this does not matter: For an ideal of finite type, the two functors coincide. But if we wish to study non-noetherian situations it might be helpful to understand the precise relation between the two definitions.)


The conjecture is not true.

By Quý’s comment, the conjecture implies that Γm is not a radical if R is a 0-dimensional local ring whose maximal ideal m is idempotent but not nilpotent. This contradicts the following result.

Lemma If R is a ring and aR is an idempotent ideal, then Γa is a radical.

Proof: Let M be an R-module, and let xM be such that x+Γa(M)Γa(M/Γa(M)). Then, axΓa(M), and therefore ax=a2x=0, implying x+Γa(M)=0.

Source : Link , Question Author : Fred Rohrer , Answer Author : Fred Rohrer

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