I'm interested in finding an algebra with elements x,y which are identified by every finitedimensional module. I'm primarily interested in the case that everything is over the complex field, but answers over other fields would also be interesting.
The Weyl algebra, generated by $x$ and $y$ subject to the single relation $xyyx=1$, is an example. A silly one, one may add: it does not have any finite dimensional representation! There are more interesting examples: let $\mathfrak g$ be the Lie algebra with basis $x_i$, $y_i$, with $i\in\mathbb Z$, and $z$, such that the only nonzero brackets are $[x_i,y_i]=z$, for all $i\in\mathbb Z$. If $A=\mathcal U(\mathfrak g)$ is the enveloping algebra of $\mathfrak g$, then $z$ acts by zero in every finite dimensional module. There are finitelygenerated examples, too. 


A similar example to Mariano's is the algebra generated by x,y subject to the relation $xy=1$. In a finite dimensional representation, yx must of course map to the same element as 1. However, there is a faithful infinite dimensional rep where x,y go to a unilateral shift and its adjoint so yx is not 1. 

