2009/5/26 Ralf Hemmecke <[email protected]>:
>
>> On Sun, May 24, 2009 at 01:44:40PM +0100, John Cremona wrote:
>>> I also would not use "ring" unless it had both a 0 and a 1.
>
>> Sorry, if I was unclearn. I was not doubting the a consensus about this.
>
> Aha, A := 3Z (all multiples of 3) is not a (mathematical) ring!? And {0}
> also doesn't count as ring?
To me, 3Z is not a ring. But {0} is the trivial ring, the only one
with 0=1. (It is therefore not a subring of any nontrivial ring since
by definition in a subring the 1 element must be thes same as in the
larger ring. It arises as the quotient R/I when I=R).
>
> That one fixes the convention that 'Ring' in any CAS to mean a
> mathematical ring with 1 is another question.
>
>> The question I was raising was about the names we wanted to use for
>> "rings" without 0 resp. 1.
>
> Z is also a ring without one, i.e., Ring should inherit from Rng. I
??? Z contained 1 last time I looked!
John
> would rather say that Rng is a "ring" that *doesn't claim* the existence
> of 1.
>
> Ralf
>
> PS: Nicolas, can you give an example of a Rg. I always thought that a
> ring is a commutative group wrt + and a semigroup wrt *.
>
> >
>
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