The notions of flatness and faithful flatness were introduced by J-P. Serre in the 1956 influential `GAGA' paper, G\'eométrie Alg\'ebrique et G\'eom\'etrie Analytique. If $R$ is a commutative ring, then an $R$-module $M$ is flat if the functor $- \otimes_R M$ on the category of $R$-modules is exact. For a flat $R$-module $M$, faithfulness amounts to demanding that $\mathfrak{m} M\subsetneq M$ for all maximal ideals $\mathfrak{m}$ in $R$. We discuss the lack of faithfulness for two flat modules: $L^2(X,\mu)$ as an $L^\infty(X,\mu)$-module (under mild assumptions), and (the Hardy Hilbert space) $H^2(\mathbb{D})$ as a (Hardy algebra) $H^\infty(\mathbb{D})$-module. On the other hand, by replacing maximal ideals $\mathfrak{m}$ by nonzero, proper, finitely-generated ideals $\mathfrak{n}$, guarantees that the analogue of faithfulness holds (that is, $\mathfrak{n}M\subsetneq M$) for the above two modules. (These facts play an important role in the ‘stabilisation problem’ in control theory, and the unfaithfulness in the $H^\infty$-$H^2$ case answers a 2005 question of Alban Quadrat.)