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Although they aren't a field as such (if I'm remembering well, because they don't form a division algebra), sedenions (16-tuples, and also further applications of the Cayley-Dickson construction) do have zero divisors (84 of them).

This makes giving any kind of sense to 0^(-1) even harder for me (when 0 is a sedenion).

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No denominators for zero divisors in the localization of any ring!!! [English: No denominators where you can multiply the denominator by anything nonzero to get zero.]

Technically, many definitions of localization do allow for zero/zero divisors to be included (we can use any multiplicatively closed set) however by the definition of localization all of the elements in our localized ring become equal to each other.

The mathematical meaning of a/b = c/d in a general ring is that there exists some value s in the set of legal denominators where s(ad - c*b) = 0. If 0 is an allowable denominator then a/b = c/d for every a,b,c,d.

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