Let $k$ be a commutative ring and let $M,N$ be two flat modules over $k$.
$\mathbf{EDIT}:$ A minimal generating set $X \subseteq M$ is a set which generates $M$ and no proper subset of $X$ generates $M$. There is no canonical notion of size for a minimal generating set, for example $\mathbb Z$ has generating sets $\{ 1 \}$ and $\{ 2, 3\}$ over $\mathbb Z$.
My question: is it true that
If $\{ m_i \mid i < \lambda\}$ is a minimal generating set for $M$ and $\{ n_j \mid j < \kappa \}$ is a minimal generating set for $N$ then $\{ m_i \otimes n_j \mid i < \lambda; j< \kappa\}$ is a minimal generating set for $M \otimes N$?
or equivalently,
If $\{ m_i \mid i < \lambda\}$ is a minimal generating set for $M$ and $\{ n_j \mid j < \kappa \}$ is a minimal generating set for $N$ then whenever a finite linear combination $ \displaystyle\sum_{(i,j)< \lambda \times \kappa}\alpha_{(i,j)} m_i \otimes n_j = 0$, each $\alpha_{(i,j)}$ is a non-unit?
I attempted to prove this for the case when $k$ is a field, thinking in terms of flatness : Let $V, W$ be vector spaces and let $\{ v_i \mid i< \lambda\}, \{ w_j \mid j < \kappa\}$ be bases for $V,W$ respectively. We can easily define maps $f \colon \oplus_{i < \lambda} k \to V$ and $g \colon \oplus_{j< \kappa} k \to W$ which send sequences $(\alpha_i )_{i< \lambda}$ to $\sum_{i < \lambda } \alpha_i v_i$ and $( \beta_j )_{j < \kappa }$ to $\sum_{j < \kappa }\beta_j w_j$. These maps are injective because of the linear independence property of the bases. By flatness the maps $ \bigoplus_{i < \lambda}k \otimes \bigoplus_{j < \kappa}k \xrightarrow{f \otimes 1} V \otimes \bigoplus_{j < \kappa}k \quad \text{and} \quad V \otimes \bigoplus_{j < \kappa} k \xrightarrow {1 \otimes g} V \otimes W $ are injective and so the composite $ \bigoplus_{(i,j) < \lambda \times \kappa} k \cong \bigoplus_{i < \lambda}k \otimes \bigoplus_{j < \kappa}k \xrightarrow {f \otimes g} V \otimes W$ is injective. This map being injective tells us that the proposed basis is actually linearly independent.
I know there are much easier ways to do that but as I said I wanted to think about the flatness of the modules more than their free-ness. However this proof does not generalise to modules because $f,g$ are not necessarily going to be injective, and the notion of linear independence doesn't really work for modules in general. Instead we have that non-unit condition as above. Is there a way to adapt this proof, or a totally different proof?