RE: Characterization of integers as a commutative ring with unit

["Stephen Lack" <S.Lack@uws.edu.au>]

Dear Andrej,

I take it by unit you mean identity (1). Then in the category of commutative
rings with 1, you presumably want the 1 to be preserved. So Z is initial, and
the trivial ring is terminal.

On the category of commutative rings without 1 (i.e. not necessarily having a 1),
there is a monoidal structure, formed by tensoring the underlying abelian groups,
and equipping this with the usual multiplication. (This would be the coproduct
in the category of commutative rings with 1, but it is not the coproduct here.)
If you allow yourself to use this extra structure, then Z is characterized as
the unit object for the tensor product.

The category of commutative rings with 1, but homomorphisms not necessarily 
preserving it, seems rather unnatural, but for what it's worth, the tensor 
product of the previous paragraph restricts to this category, and so can be
used to characterize Z once again.

Regards,

Steve Lack.


-----Original Message-----
From: cat-dist@mta.ca on behalf of Andrej Bauer
Sent: Thu 10/26/2006 6:56 PM
To: categories@mta.ca
Subject: categories: Characterization of integers as a commutative ring with unit
 
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