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Likewise multiplication has axioms of commutativity, associativity, an identity element different from the additive identity element, but not claimed to be unique. There's distributivity over addition. Then there's reciprocity, for all elements except the addition identity.

I hate exceptions! Is the exclusion absolutely necessary, or is it possible to have some kind of number system where the additive identity element can have a multiplicative reciprocal? On page 4 in Shilov's book, it says that the distributive property connects the operation of multiplication with addition, but isn't the distinctiveness of the two operator's identity elements and the reciprocal exclusion of zero a more forceful connection? Perhaps it's the same thing.