Let $R$ be a noetherian local ring and let $M$ be a finite $R$-module. Assume that the annihilator of $M$ is zero. Consider a minimal presentation of M as follows: $R^n\stackrel{\varphi}{\longrightarrow}R^m\longrightarrow M\longrightarrow0$. Can we conclude that $m>n$, or is it also possible to have $m\leq n$ with all $m\times m$ minors of the presentation matrix $\varphi$ equal to zero?



Presentation of finite modules with null annihilator
Graham's comment gave some simple counterexamples. I will show that even if $R$ is nice, say a Gorenstein domain, there will always be a lot of counter-examples. 


Let $M$ be a non-free maximal CM module over $R$. Consider a minimal presentation:
$$ 0 \to N \to R^n \to R^m \to M \to 0 $$
If $m\leq n$ we found our counter example. If $m>n$ then dualizing the sequence (note that since $R$ is Gorenstein dualizing preserve exactness), so one gets a sequence:


$$ 0 \to M^\* \to R^m \to R^n \to N^\* \to 0 $$


hence $N^\*$ is a counter-example! 


