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To elaborate on Karl's comment:

Let $m$ be the irrelevant ideal of $R$, then there is a short exact sequence:

$$0 \to H_m^0(M) \to M \to \Gamma^*(\mathcal{F}) \to H_m^1(M) \to 0$$

(see Eisenbud's book, Theorem A4.1, p. 693). Here $H_m^i(M)$ denote the local cohomology modules. So the map is injective precisely when $H_m^0(M):= \cup_n (0:_M m^n) = 0$. I believe such module is called $m$-torsion-free (don't know a reference off hand, may be Brodman-Sharp's Brodmann-Sharp's book on local cohomology?).

Also, it is equivalent to $m$ contains a non-zerodivisor on $M$ (may be that's what you meant in the last paragraph?)
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To elaborate on Karl's comment:

Let $m$ be the irrelevant ideal of $R$, then there is a short exact sequence:

$$0 \to H_m^0(M) \to M \to \Gamma^*(\mathcal{F}) \to H_m^1(M) \to 0$$

(see Eisenbud's book, Theorem A4.1, p. 693). Here $H_m^i(M)$ denote the local cohomology modules. So the map is injective precisely when $H_m^0(M):= \cup_n (0:_M m^n) = 0$. I believe such module is called $m$-torsion-free (don't know a reference off hand, may be Brodman-Sharp's book on local cohomology?cohomology?).

Also, it is equivalent to $m$ contains a non-zerodivisor on $M$ (may be that's what you meant in the last paragraph?)
show/hide this revision's text 1

To elaborate on Karl's comment:

Let $m$ be the irrelevant ideal of $R$, then there is a short exact sequence:

$$0 \to H_m^0(M) \to M \to \Gamma^*(\mathcal{F}) \to H_m^1(M) \to 0$$

(see Eisenbud's book, Theorem A4.1, p. 693). Here $H_m^i(M)$ denote the local cohomology modules. So the map is injective precisely when $H_m^0(M):= \cup_n (0:_M m^n) = 0$. I believe such module is called $m$-torsion-free (don't know a reference off hand, may be Brodman-Sharp's book on local cohomology?)