Let $U$ and $V$ be modules over a commutative ring $K$. Is the canonical map $U^* \otimes V^* \to (U \otimes V)^*$ always injective?
I'm a differential geometer so I'm usually dealing with finite-rank locally free sheaves of modules where this is an isomorphism---I'm not very familiar with the weirdness that can happen in more general modules over other commutative rings.