From: Josh Nichols-Barrer <jnb@math.mit.edu>
To: categories@mta.ca
Subject: Re: Characterization of integers as a commutative ring with unit
Date: Thu, 26 Oct 2006 21:09:40 -0400 (EDT) [thread overview]
Message-ID: <E1GdQuV-0004km-PS@mailserv.mta.ca> (raw)
Hi Andrej,
Isn't Z the initial /ring/? 0 isn't initial, as 0=1 holds only in itself
(Spec Z is the terminal scheme, Spec 0 the empty scheme, so the initial
scheme).
-Josh
On Thu, 26 Oct 2006, Andrej Bauer wrote:
> For the purposes of defining the data structure of integers in a
> Coq-like system, I am looking for an _algebraic_ characterization of
> integers Z as a commutative ring with unit. (The one-element ring is a
> ring.)
>
> Some possible characterizations which I don't much like:
>
> 1) Z is the free group generated by one generator. I want the ring
> structure, not the group structure.
>
> 2) Z is the free ring generated by the semiring of natural numbers. This
> just translates the problem to characterization of the semiring of
> natural numbers.
>
> 3) Z is the initial non-trivial ring. This is no good because
> "non-trivial" is an inequality "0 =/= 1" rather than an equality.
>
> 4) Let R be the free commutative ring with unit generated by X. Then Z
> is the image of the homomorphism R --> R which maps X to 0. This is just
> ugly and there must be something better.
>
> I feel like I am missing something obvious. Surely Z appears as a
> prominent member of the category of commutative rings with unit, does it
> not?
>
> Best regards,
>
> Andrej
>
>
next reply other threads:[~2006-10-27 1:09 UTC|newest]
Thread overview: 9+ messages / expand[flat|nested] mbox.gz Atom feed top
2006-10-27 1:09 Josh Nichols-Barrer [this message]
-- strict thread matches above, loose matches on Subject: below --
2006-10-27 9:29 George Janelidze
2006-10-27 7:51 Stephen Lack
2006-10-27 7:23 George Janelidze
2006-10-27 0:01 Stephen Lack
2006-10-26 20:26 Andrej Bauer
2006-10-26 15:21 Fred E.J.Linton
2006-10-26 14:48 Steve Vickers
2006-10-26 8:56 Andrej Bauer
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