George M. Bergman, Zero-divisors in tensor products

pp.32-82 of SLNM v.545, Noncommutative ring theory, papers presented at the international conference at Kent State University, 4-5 April 1975, Springer Verlag, 1976. MR 55#12752 . 

Louis Rowen and David Saltman have a 2011 preprint, Tensor Products of Division Algebras and Fields, 17pp., in which they answer the main question I was struggling with.  They construct division algebras  D1  and  D2,  each finite-dimensional over its center, whose centers contain a common algebraically closed field  F,  such that  D1 ⊗F D2  has zero-divisors.

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