Homogeneous non zero divisor in a graded module.

Question

Let $(R_0,m_0)$ be a local ring and $R=\oplus_{n\geq 0}R_n$ a positively standard graded ring.
Let $M$ be a finitely generated graded module over $R$.
Let $R_+$ be the ideal $\oplus_{n>0}R_n$ and $\Gamma_{R_+}(M)$ the $R$-submodule of $M$ given by $\{m\in M\mid\exists n\in \mathbb{N}\text{ such that }R_+^nm=0\}$.
How can I prove that exists "$x$" in $R$ homogeneous and not zero divisor over $\bar{M}:=\frac{M}{\Gamma_{R_+}(M)}$?

I know that $R$ contains a not zero divisor over $\bar{M} $ but i don't know how to deduce that it can be taken homogeneous.
Can someone please help me?
Thanks

Answer 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 $\bar M$.

Answer 2

I have a solution:
As $\bar{M}:=\frac{M}{\Gamma_{R_+}(M)}$ satisfies $\Gamma_{R_+}(\bar{M})=0$ one has that $0:_{\bar{M}}R_+={0}$ so $R_+\nsubseteq\cup_{p\in Ass(\bar{M})}p$ now it suffice to apply a graded version of prime avoidance to conclude that exists $x\in R_+$ which is not a zerodivisor over $\bar{M}$.

Comment: I did't know that there was a version of prime avoidance lemma, as a reference i look at https://commalg.subwiki.org/wiki/Prime_avoidance_lemma.