Re: Characterization of integers as a commutative ring with unit

Re: Characterization of integers as a commutative ring with unit

[George Janelidze]

Dear Steve,

In your message of October 27 to Andrej Bauer you say:

"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."

But no, this is not an extra structure, which, explained properly, has some
obvious and some non-obvious aspects - see [A. Carboni and G. Janelidze,
Smash product of pointed objects in lextensive categories, Journal of Pure
and Applied Algebra 183, 2003, 27-43] (I also gave a talk about this called
"Abstract commutative algebra I: Associativity of tensor (=co-smash)
products (12.12.2001)" on Australian Category Seminar).

In simple words: tensor product of commutative rings of A and B without 1 is
nothing but their co-smash product (=the kernel of the canonical morphism
A+B ---> AxB), and therefore Z is the unit object of the smash product. This
observation itself might be infinitely old - simply because it is simple!
But the reason of the associativity of the smash product and the very
definition of associativity is a different story (e.g. the associativity
isomorphism itself is not an extra structure as it happens in a general
monoidal category).

Let me also point out that the co-smash product is to be investigated in any
semi-abelian category. Note that:

In any semi-abelian category the canonical morphism A+B ---> AxB is a
regular=normal epimorphism for each two objects A and B. Therefore the
co-smash product of A and B is not merely its kernel - IT IS THE MEASURE OF
NONADDITIVITY. And you can define an abelian category as a semi-abelian
category with trivial co-smash products. In this sense the category CR of
commutative rings without 1 is "very nonabelian" - since instead of having
trivial co-smash products it has a unit object for the co-smash product
(this is like a monoid with zero versus a semigroup with zero and zero
multiplication). On the other hand this makes CR "almost abelian" since it
is one of the very few semi-abelian categories where the co-smash product is
associative (it is not the case for groups, not for non-commutative rings,
etc.).

George Janelidze

Re: Characterization of integers as a commutative ring with unit

[George Janelidze]

Dear Steve,

Thank you for your kind words. I have mentioned my paper with Aurelio only
because there are too many details that I could not describe in a brief
email message. But if it comes to "...I should have mentioned explicitly
your work with Aurelio...", I can say the same about myself: I should have
mentioned your (very important!) paper

[A. Carboni, S. Lack, and R. F. C. Walters, Introduction to extensive and
distributive categories, Journal of Pure and Applied Algebra 84, 1993,
145-158]

and Bill Lawvere's original question about commutative rings and many other
things that Aurelio did mention in his CT1999 talk. (In any case, I hope you
do not assume that I know what "Coq" is, do you?)

Yours-

George

----- Original Message -----
From: "Stephen Lack" <S.Lack@uws.edu.au>
To: "George Janelidze" <janelg@telkomsa.net>; <categories@mta.ca>
Cc: "Andrej Bauer" <Andrej.Bauer@fmf.uni-lj.si>
Sent: Friday, October 27, 2006 9:51 AM
Subject: RE: categories: RE: Characterization of integers as a commutative
ring with unit


Dear George,

I remember well your lovely talk and paper, and indeed I had this
in mind when I wrote. I shouldn't have used the words "extra structure"
(in fact I said this really because I don't know what "Coq" is) and I
should have mentioned explicitly your work with Aurelio.

Sorry.

Steve.

RE: Characterization of integers as a commutative ring with unit

[Stephen Lack]

Dear George,

I remember well your lovely talk and paper, and indeed I had this
in mind when I wrote. I shouldn't have used the words "extra structure"
(in fact I said this really because I don't know what "Coq" is) and I 
should have mentioned explicitly your work with Aurelio.

Sorry.

Steve.



-----Original Message-----
From: George Janelidze [mailto:janelg@telkomsa.net]
Sent: Fri 10/27/2006 5:23 PM
To: categories@mta.ca; Stephen Lack
Cc: Andrej Bauer
Subject: Re: categories: RE: Characterization of integers as a commutative ring with unit
 
Dear Steve,

In your message of October 27 to Andrej Bauer you say:

"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."

But no, this is not an extra structure, which, explained properly, has some
obvious and some non-obvious aspects - see [A. Carboni and G. Janelidze,
Smash product of pointed objects in lextensive categories, Journal of Pure
and Applied Algebra 183, 2003, 27-43] (I also gave a talk about this called
"Abstract commutative algebra I: Associativity of tensor (=co-smash)
products (12.12.2001)" on Australian Category Seminar).

In simple words: tensor product of commutative rings of A and B without 1 is
nothing but their co-smash product (=the kernel of the canonical morphism
A+B ---> AxB), and therefore Z is the unit object of the smash product. This
observation itself might be infinitely old - simply because it is simple!
But the reason of the associativity of the smash product and the very
definition of associativity is a different story (e.g. the associativity
isomorphism itself is not an extra structure as it happens in a general
monoidal category).

Let me also point out that the co-smash product is to be investigated in any
semi-abelian category. Note that:

In any semi-abelian category the canonical morphism A+B ---> AxB is a
regular=normal epimorphism for each two objects A and B. Therefore the
co-smash product of A and B is not merely its kernel - IT IS THE MEASURE OF
NONADDITIVITY. And you can define an abelian category as a semi-abelian
category with trivial co-smash products. In this sense the category CR of
commutative rings without 1 is "very nonabelian" - since instead of having
trivial co-smash products it has a unit object for the co-smash product
(this is like a monoid with zero versus a semigroup with zero and zero
multiplication). On the other hand this makes CR "almost abelian" since it
is one of the very few semi-abelian categories where the co-smash product is
associative (it is not the case for groups, not for non-commutative rings,
etc.).

George Janelidze

Re: Characterization of integers as a commutative ring with unit

[Josh Nichols-Barrer]

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
>
>

RE: Characterization of integers as a commutative ring with unit

[Stephen Lack]

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

Re: Characterization of integers as a commutative ring with unit

[Andrej Bauer]

I would like to thank all 11 people who pointed out that Z _is_ the
initial unital ring. I forgot to take into account that homomorphisms of
unital rings map 1 to 1, so the trivial ring is not initial (and Z is).
Thank you for being kind even though I asked a trivial questions.

Andrej

Re: Characterization of integers as a commutative ring with unit

[Fred E.J.Linton]

What is not to like about Z being the initial ring-with-unit? No need to
impose 0 {/ne} 1 -- it follows.

Cheers,

-- Fred.

Re: Characterization of integers as a commutative ring with unit

[Steve Vickers]

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
>
>

Characterization of integers as a commutative ring with unit

[Andrej Bauer]

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