A few years ago I gave a departmental colloquium talk, aimed at beginning M.A. students, on "An application of calculus to ring theory."  A slightly facetious little abstract can be found here.  The example establishing the main results—very well known to workers in commutative ring theory—was the ring of germs at $0$ of class $C^{\infty}$ functions on $\mathbb{R}$.  A bit of calculus is needed in verifying the requisite properties.