## Reference request: conditions for the kernel of a linear map from $\mathbb{Z}_m^n \to \mathbb{Z}_m^k$ to be free

Let $$\mathbb{Z}_m = \mathbb{Z}/m\mathbb{Z}$$. Let $$A$$ be an $$k \times n$$ matrix. Let $$f: \mathbb{Z}_m^n \to \mathbb{Z}_m^k$$ be a linear map defined by $$f(x) = Ax$$, $$x \in \mathbb{Z}_m^n$$. Are there some references about the condition of $$\ker(f)$$ to be a free $$\mathbb{Z}_m$$-module?

For example, the kernel of the map $$f: \mathbb{Z}_4 \to \mathbb{Z}_4$$ given by $$f(x) = 2x$$ is $$\ker(f) = \{0, 2\}$$. Therefore $$\ker(f)$$ is not free.

Let $$f: \mathbb{Z}_m^4 \to \mathbb{Z}_m^2$$ be a linear map given by $$f(x) = (2x_1-x_3-x_4, x_2-x_3-x_4)^T$$. Then $$\ker(f) = \mathbb{Z}_m^2$$. In this case, $$\ker(f)$$ is free. Thank you very much.