I am currently self studying the textbook Commutative Algebra with a View Toward Algebraic Geometry by David Eisenbud and came across the following exercise:
Exercise 6.10 (Flatness of graded modules): Let $R=R_0\oplus R_1\oplus\ldots$ be a graded ring with $R_0$ a field, and let $M$ be a graded $R$-module. Show that $M$ is flat over $R$ iff the multiplication map $I\otimes_R M\rightarrow M$ is an injection for every homogeneous ideal $I$ of $R$.
What I've tried: Let $I$ be an ideal of $R$ and $J$ the ideal generated by the homogeneous forms of $I$. Suppose some $x=\sum_ir_i\otimes m_i\in I\otimes_R M$ goes to zero in $M$. Thus $\sum_ir_im_i=0$. Since $M$ has a homogeneous generating set, we can assume that the $m_i$ are homogeneous. Let $r_{ij}$ be the homogeneous component of $r_i$ of $j$-th degree. Splitting each $r_i$ in the sum above into homogeneous components gives $\sum_i\sum_jr_{ij}m_i=0$. We can rearrange this sum into its homogeneous components, which gives $\sum_k\sum_{\deg(r_{ij}m_i)=k}r_{ij}m_i=0$ where we are summing over all $i,j$ such that $\deg(r_{ij}m_i)=k$. Since each component has a different degree, it follows that for every $k$, $\sum_{\deg(r_{ij}m_i)=k}r_{ij}m_i=0$. By the hypothesis, the element $\sum_{\deg(r_{ij}m_i)=k}r_{ij}\otimes m_i=0$ in $J\otimes_R M$. Let $\mathbf{r}_k$ be the vector whose elements are the $r_{ij}$ satisfying $\deg(r_{ij}m_i)=k$ and $\mathbf{m}_k$ the vector of the same length whose elements are the $m_i$ satisfying $deg(r_{ij}m_i)=k$ so that $\sum_{\deg(r_{ij}m_i)=k}r_{ij}m_i=\mathbf{r}_k^\top\mathbf{m}_k$. Suppose that $\mathbf{r}_k$ has length $p$. According to Lemma 6.4 in Eisenbud, there exists a positive integer $q$, elements $m'_1,\ldots,m_q'\in M$ (which we label as a vector $\mathbf{m}'_k$) and a matrix $A_k\in M_{p\times q}(R)$ such that $\mathbf{r}_k^\top A_k=0$ and $A_k\mathbf{m}'_k=\mathbf{m}_k$.
At this point I don't know how to continue. In principle one wants to somehow combine the matrices $A_k$ in such a way to get a similar condition on the elements $r_i$ and $m_i$ to be able to show that $x=0$, however I am clueless as to how to do this. Am I missing something or is there perhaps a better way to solve this problem?