A Dedekind-Mertens theorem for power series rings

We prove a power series ring analogue of the Dedekind-Mertens lemma. Along the way, we give limiting counterexamples, we note an application to integrality, and we correct an error in the literature.
Comment: Added two references, as per referee's request. 9 pages, to appear in Proc. Amer. Math. Soc