Whether a non-zero module can have a zero socle? Let $M$ be a module. Then $soc(M)=\sum\{N\leq M| \text{$N$ is a simple submodule of $M$}\}=\cap\{L\leq M| \text{$L$ is essential in $M$}\} $. I don't know whether a non-zero module can have zero socle? Who can give me an example?

In the context of finite-dimensional representations, you always have a nonzero socle. Hint: Consider subrepresentations of minimal dimension. In general, it's not true. Hint: Consider torsion-free modules over integral domains, e.g. over $k[x]$. 
