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.)

Answer

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.

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

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