Abstract
We construct Hilbert C^*-modules useful for studying Gabor systems and show that they are Banach algebras under pointwise multiplication. For rational ab<1 we prove that the set of functions g \in L^2(R) so that (g,a,b) is a Bessel system is an ideal for the Hilbert C^*-module given this pointwise algebraic structure. This allows us to give a multiplicative perturbation theorem for frames. Finally we show that a system (g,a,b) yields a frame for L^2(R) iff it is a modular frame for the given Hilbert C^*-module.