I am currently reading the chapter on finitely-generated modules over PIDs in Jacobson's Basic Algebra and at several points he uses the property that if the annihilator of an element $x$ of the module (i.e. the elements $r$ of the base ring $R$ s.t. $rx=0$) is the entire base ring $R$, then the element is zero (i.e. $\text{ann}(x)=R\Rightarrow x=0$). I seem to have missed the part where he actually proves this or states why it is true, so I would like to ask what the reason for this might be in this specific case (finitely-generated modules over PIDs) and whether it holds in more general cases (non-finitely-generated modules over more general rings).
Thank you in advance.
Best Answer
It's enough to assume that the ring has identity element (and the module is unitary: $1x=x$).
Then in particular for $1\in R$ we get $0=1x=x$.