Tensor Products and Computability

We consider the problem of computability in tensor products of modules over a ring. We exhibit a finite local ring A and a pair of A-modules, given explicitly by generators and relations, with the following property. The operations in each module are computable in polynomial time, but equality in their tensor product is undecideable. The construction is of interest because it directly embeds Turing machine-like computations into the tensor product. We also present sufficient conditions for equality in tensor products to be decideable.