(I removed my motivation because it may be misleading :) )

Let $A$ be a noetherian commutative ring and let $M \neq 0$ be a finitely generated zero-dimensional (i.e. $\mathrm{dim} \ \mathrm{Supp}(M) = 0$) $A$-module. Then the submodule $0 < M$ has primary decomposition $0 = \bigcap_{\mathfrak{p} \in \mathrm{Supp}(M)} M(\mathfrak{p})$, where $M(\mathfrak{p})$ is the $\mathfrak{p}$-primary component of $M$, i.e. the kernel of the canonical morphism $M \rightarrow M_{\mathfrak{p}}$. I have proven (it's hopefully correct) that the canonical morphism $\mathbf{j}:M \rightarrow \bigoplus_{\mathfrak{p} \in \mathrm{Supp}(M)} M/M(\mathfrak{p})$ ist an isomorphism and that $\mathbf{j}(M \lbrack \mathfrak{q}^\infty \rbrack) \leq M/M(\mathfrak{q}) \leq \bigoplus_{\mathfrak{p} \in \mathrm{Supp}(M)} M/M(\mathfrak{p})$, where $M \lbrack \mathfrak{q}^\infty \rbrack$ is the $\mathfrak{q}$-torsion part of $M$ ($0$-th local cohomology with support $\mathfrak{q}$). Now, my question is if in general $\mathbf{j}(M \lbrack \mathfrak{q}^\infty \rbrack) = M/M(\mathfrak{q})$ so that $M = \bigoplus_{\mathfrak{p} \in \mathrm{Supp}(M)} M \lbrack \mathfrak{p}^\infty \rbrack$, or do I need to assume that $A$ is a Dedekind ring (and perhaps also that $M$ is a torsion module) and if so, how can I prove this?