Let $\mathfrak{g}=\mathfrak{sl}_2(\mathbb{C})$. I'm reading a proof that a $\mathfrak{g}$-module is torsion-free iff its socle is torsion-free.
One direction is clear, but for the other, if $M$ has torsion, then one gets that $M$ has a nonzero weight vector $v$. They say that the submodule generated by this weight vector intersects the socle nontrivially, so the socle is not torsion-free.
Why does the weight submodule have to intersect the socle nontrivially? That's the part I don't understand.