  Closed essential surfaces in a three-manifold can be detected by ideal points
of the character variety or by algebraic non-integral representations. We give
examples of closed essential surfaces not detected in either of these ways. For
ideal points, we use Chesebro's module-theoretic interpretation of
Culler-Shalen theory. As a corollary, we construct an infinite family of closed
hyperbolic Haken 3-manifolds with no algebraic non-integral representations
into PSL(2, C), resolving a question of Shanuel and Zhang.
