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Rewrite of article

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The article was little more than a stub, and gave no references. It also contained at least one error, and some of the information it gave was arguably not in the most helpful form. I have rewritten it from scratch to give a fuller account.JamesBWatson (talk) 17:24, 1 December 2008 (UTC)[reply]

Equivalence of axioms

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Can you elaborate on why "A2, A4, A5" can't be replaced with "A4*" in the definition? It seems clear to me that this can be done. Arun (talk) 22:32, 26 August 2009 (UTC)[reply]

If anyone else is interested in this, it occurs to me now that a much simpler answer than the one I originally gave (below) is this: Because A4* tells us nothing about multiplication by zero, whereas both A2 and A4 do so. JamesBWatson (talk) 10:58, 21 January 2010 (UTC)[reply]
Unless I have missed something, this is already answered in the article. The "exceptional structure" defined in the last sentence of "Notes on the definition" number 2 satisfies A1, A3, and A4* but clearly not A4 (consider ).
As for A2 (associativity of multiplication), it is independent of all the other axioms, as can be seen from the following example, which obeys all the axioms except A2. Take a field of order 4. Let i be an element of the field which is not equal to 0 or 1. Change the definition of multiplication by setting both i.0 and (1+i).0 equal to 1 instead of 0.
JamesBWatson (talk) 19:34, 31 August 2009 (UTC)[reply]