Let $k$ be a finite field. Let $k[x] \subseteq R$ where $R$ is a noetherian, onedimensional and reduced ring such that $R$ is finite free over $k[x]$ (in particular, the polynomials in $x$ are no zero divisors in $R$).
Could you give me an example of a set $Z \subseteq R$ with $\lvert Z\rvert \geq 2$ such that $Z$ is $k[x]$-linearly independent and, moreover, every $k$-linear combination of its elements is a zero divisor?
Let $A=k[y], B=k[z]$ and let $R=A\times B$. We have an inclusion $k[x]\subset R$ where $x\mapsto (y^2,z^2)$. Then, $R$ is free over $k[x]$ with basis $(1,0), (0,1), (y,0), (0,z)$. Take $Z=\{(1,0), (y,0)\}$. Then every $k[x]$ linear combination of elements of $Z$ is a zero divisor, not just $k$-linear combinations. Every one such is killed by the non-zero element $(0,1)$.