Rotwang
Posts:
1,679
From:
Swansea
Registered:
7/26/06


Re: Can addition be defined in terms of multiplication?
Posted:
Aug 16, 2013 6:41 PM


On 16/08/2013 17:14, Helmut Richter wrote: > [...] > > A similar problem I have asked some years ago is the following: > > Given a multiplication on a set (e.g. defined as a commutative and > associative operation allowing cancellation (ab = ac implies b = c)), > is there an addition so that the set becomes a ring with both operations? > I have no clue how to tackle such questions. > > An example: Let M = {x elem Z : x == 1 mod 3} with ordinary > multiplication. Could this be the multiplication in a ring, if addition is > suitably defined? I guess, no, but it is but a guess.
I think the answer is no. Note that in any ring,
x.0 = x.0 + (x.x  x.x) = x.(0 + x)  x.x = x.x  x.x = 0
So if M with ordinary multiplication could be extended to a ring, there would be an element of M which multiplied by every other element to give itself. No such element exists (since if e.g. x.4 = x then x = 0).

