Homogeneous non zero divisor in a graded module.
Solution 1:
In fact, one knows that $R_+$ contains a non-zerodivisor on $\bar M$ since the later is $R_+$-torsion-free. In particular, $R_+$ is not contained in the union of the associated primes of $\bar M$. But the associated primes are homogeneous, and by the homogeneous prime avoidance $R_+$ contains a homogeneous element which does not belong to any associated prime of $\bar M$, so it is a non-zerodivisor on $\bar M$.