Re: Characterization of integers as a commutative ring with unit

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

Dear Andrej,

Z is the initial ring with unit. (Doesn't matter whether you require
commutativity.)

It's not clear to me why you felt the need to say "non-trivial" in (3).

Regards,

Steve.

On 26 Oct 2006, at 09:56, 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
>
>