- No.

$A := \{(i,j) \in \mathbb{N}^2, j\leq i\}$

$S := \{$finite subsets of A with at least 2 different first coordinates$\}$

$R := \mathbb{F}_2[t_s: s \in S]/(t_{s_1}t_{s_2}: s_1,s_2 \in S)$

$M := \oplus_{(i,j) \in A}Rm_{ij}/(t_s\Sigma_{(i,j)\in s}m_{ij}: s \in S)$

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

$\{m_{m1},..m_{mm}\} \subset M$ is a linearly independent set of size $m$.

Let $T := (t_s: s \in S)R$, non-zero ideal of $R$.

Suppose $\{x_k\} \subset M$ is a linearly independent set.

- Write $x_k = \Sigma r_{ij}m_{ij}$. Then $\{r_{ij}(0)\} \neq \{0\}$: 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 $\{x_k\} \subset \Sigma_nRm_{mn}$ mod $T$: otherwise $\cup_k\{(i,j):r_{ij}(0) \neq 0\}$ has at least 2 different first coordinates and some $t_s$ kills some $x_k$ or some $x_{k_1}+x_{k_2}$.
- $\#\{x_k\} \leq m: dim_{\mathbb{F}_2}\Sigma_nRm_{mn}$ mod $T = m$.