I know what a non-degenerate bi-linear form is, but what does it mean for say a left $R$-module $M$ to be non-degenerate? (Here $R$ is a ring without unit$)
I came across a module being called non-degenerate studying representation theory.
Here is the definition which I have seen in Cartier's article on the Representation theory of p-adic groups (it appears in the Corvallis proceedings):
This is proving amazingly resistant to an internet search.
A few of the first hits left me with the impression it might just mean that the map $$M\times R\rightarrow M$$ is a nondegenerate bilinear map.
Edit: The link in the comments for this solution say something of this sort. They say "if $xR=0$ for an $x$ in the module, then $x=0$" I think it is mainly meant to guarantee the the annihilator of an element isn't the whole ring (having an identity normally precludes that.)