Question 1. has a negative answer.

Denote $A := \{(i,j) \in \mathbb{N}^2, j\leq i\}$ and $S := \{$finite subsets of A with at least 2 different first coordinates$\}$. Define the ring $$R := \mathbb{F}_2[t_s: s \in S]\,/\,\big(t_{s_1}t_{s_2}: s_1,s_2 \in S\big),$$ and the $R$-module
$$M := \bigoplus_{(i,j) \in A}Rm_{ij}\,/\,\left(t_s\sum_{(i,j)\in s}m_{ij}: s \in S\right).$$

For example, $s_0=\{(1,1),(2,1))\} \in S$ and $t_{s_0}(m_{11}+m_{21})=0.$

Then for each $m$, the subset $\{m_{m1},..m_{mm}\} \subset M$ is a linearly independent subset of size $m$.

Suppose that $(x_k)_{k\in K} \subset M$ is a linearly independent family and let us show that $K$ is finite.
Denote $T := (t_s: s \in S)R$; this is a non-zero ideal of $R$.

- Write $x_k = \Sigma r_{ij}^{(k)}m_{ij}$. Then $\{r_{ij}^{(k)}(0)\} \neq \{0\}$: otherwise $Tx_k=0$.
- $\{x_k\}$ is linearly independent mod $T: \Sigma r_kx_k \in TM \Rightarrow \Sigma t_{s_0}r_kx_k =0 \Rightarrow t_{s_0}r_k=0$ all $k \Rightarrow r_k \in T$ all $k$.
- $\exists m$ such that $\{x_k\} \subset \sum_nRm_{mn}$ mod $T$: otherwise $\bigcup_k\{(i,j):r_{ij}(0) \neq 0\}$ has at least 2 different first coordinates and some $t_s$ kills some $x_k$ or some $x_{k_1}+x_{k_2}$.
- $\#(K) \leq m: \dim_{\mathbb{F}_2}\sum_nRm_{mn}$ mod $T = m$.

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