Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 Helmut Richter Posts: 164 Registered: 7/4/06
Re: Can addition be defined in terms of multiplication?
Posted: Aug 17, 2013 5:11 PM

On Fri, 16 Aug 2013, Shmuel (Seymour J.) Metz wrote:

> 08/16/2013
> at 06:14 PM, Helmut Richter <hhr-m@web.de> said:
>

> >Given a multiplication on a set (e.g. defined as a commutative
> >and associative operation allowing cancellation (ab = ac implies
> >b=c)),

>
> That would be a strange definition in general, although it's
> reasonable if you're only concerned with groups. Is the "*" operation
> in Z a multiplication? Certainly 0b=0c doesn't imply b=c.

I was imprecise in stating the problem. Of course, it was meant in a way so
that it *can* have solutions, e.g. the ordinary multiplication of integers is
indeed the multiplication in a ring. Let me try to do better:

Given a structure (M, 0, 1, ·) where

M is a set,
0 elem M is a constant,
1 elem M is a constant,
· : M×M -> M is an operation

with the following properties:

(1) · is commutative and associative
(2) 0·x = 0 for all x
(3) 1·x = x for all x
(4) x·z = y·z and z ¬= 0 ==> x = y for all x, y, z

where condition (4) comes from the first intended application of the problem;
the problem may be meaningful also without it. Condition (4) is not generally
fulfilled in rings.

Now we want to decide the question whether there is an operation + so that
(M, 0, 1, +, ·) is a ring.

Example 1: M = {0, 1, 2, 3, 4}
If x, y ¬= 0, x·y is the unique z in {1, 2, 3, 4} such that
x+y-1 == z (mod 4)

Then there is an addition which makes it a ring, even a field:
1+1=2; 1+3=0; 1+4=3; 2+2=3; 2+3=1; 2+4=0; 3+3=4; 3+4=2; 4+4=1;

because {1, 2, 3, 4} is a group under · that is isomorphic to the
multiplicative group of F5, and the isomorphism consists of swapping 3 and 4.

Example 2: M = {0} u {x elem Z: x == 1 (mod 3)}
x·y is defined as ordinary multiplication

Question: Is there an addition which makes that a ring? Or why not?

--
Helmut Richter

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