When Ideal transform functor is Artinian?

Question

Let $R$ be a Noetherian ring and $I$ an ideal. For a finitely generated module $M$ over $R$ we define $$D_I(M)=\varinjlim_\limits{n\geq1}\operatorname{Hom}_R(I^n,M).$$

A variety of nice results about $D_I(M)$ were shown in "Local cohomology. An algebraic introduction with geometric applications".

Now let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a non-zero finitely generated $R$-module. When $D_{\mathfrak{m}}(M)$ is artinian over $R$?

Answer 1

In one case, if $M$ is artinian itself it follows from exact sequence $$0\longrightarrow \frac{M}{\Gamma_I(M)}\longrightarrow D_I(M)\longrightarrow H^1_I(M)\longrightarrow 0$$ that $D_I(M)$ is artinian, because $H^1_I(M)$ is artinian.