Virgil
Posts:
8,833
Registered:
1/6/11


Re: Can addition be defined in terms of multiplication?
Posted:
Aug 16, 2013 8:01 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.
In rings, ab = ac need not necessarily imply b = c.
It requires, in addition, at least that both ab and ac be nonzero.
Note that in the ring of integers modulo 16 one has 8*2 = 8*4 = 0 but not 2 = 4 

