Let $ f$ be an irreducible polynomial of degree $q$ over $\mathbb{F}_p$. Let ${\bf F}=\frac{\mathbb{F}_p[x]}{f}$ be the finite field which contain $p^q$ elements. Assume $k>1$ is an integer and suppose that ${\bf R}=\frac{\mathbb{F}_p[x]}{f^k}$ is the quotient ring such that the coefficients come from $\mathbb{F}_p$ and the multiplication of elements are reduced by $f^k$.

Let $\bf A$ be an $n \times m$ matrix such that the elements of $\bf A$ are polynomials over $\mathbb{F}_p[x]$. Suppose that the minimum number of linearly dependent columns of the matrix $\bf A$ over ${\bf F}$ and ${\bf R}$ are denoted with ${\operatorname{MD}}_F$ and ${\operatorname{MD}}_R$, respectively.

**My question:**

How to prove that if ${\operatorname{MD}}_F=r$ for some positive integer $1\leq r \leq m$, then ${\operatorname{MD}}_R \geq r$.

Furthermore, Is it possible to make some conditions over $\bf A$ such that we have ${\operatorname{MD}}_F=r$ and ${\operatorname{MD}}_R > r$.

In practical, I work with $p=2$.

*Thanks for any help.*