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Topic: Can addition be defined in terms of multiplication?
Replies: 58   Last Post: Aug 23, 2013 3:56 PM

 Messages: [ Previous | Next ]
 Rotwang Posts: 1,685 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).

Date Subject Author
8/16/13 Peter Percival
8/16/13 William Elliot
8/16/13 Peter Percival
8/16/13 David C. Ullrich
8/16/13 namducnguyen
8/17/13 Peter Percival
8/17/13 namducnguyen
8/17/13 fom
8/23/13 tommy1729_
8/16/13 Peter Percival
8/16/13 Robin Chapman
8/16/13 Helmut Richter
8/16/13 Rotwang
8/16/13 Virgil
8/22/13 Rock Brentwood
8/16/13 Shmuel (Seymour J.) Metz
8/17/13 Helmut Richter
8/16/13 Jim Burns
8/16/13 fom
8/17/13 Robin Chapman
8/17/13 fom
8/17/13 Peter Percival
8/17/13 fom
8/17/13 Peter Percival
8/17/13 Peter Percival
8/18/13 William Elliot
8/18/13 Peter Percival
8/18/13 William Elliot
8/18/13 Peter Percival
8/18/13 Graham Cooper
8/18/13 David C. Ullrich
8/18/13 David C. Ullrich
8/17/13 Graham Cooper
8/18/13 David Bernier
8/18/13 Ben Bacarisse
8/18/13 Peter Percival
8/18/13 Jim Burns
8/18/13 fom
8/18/13 Ben Bacarisse
8/18/13 Graham Cooper
8/18/13 Graham Cooper
8/18/13 Graham Cooper
8/18/13 Graham Cooper
8/19/13 Graham Cooper
8/19/13 Alan Smaill
8/19/13 fom
8/19/13 Alan Smaill
8/20/13 Alan Smaill
8/20/13 Peter Percival
8/20/13 Graham Cooper
8/20/13 Graham Cooper
8/22/13 David Libert
8/22/13 Peter Percival
8/20/13 fom