Definition of a field

[fl]

In the definition of a field there is the condition that the
underlying ring is not the
zero ring. This condition puzzles me. What is it for? Is it an
important condition
or is it of those surnumerary conditions that often appears in
definitions?

Note: here is the example of a textbook where this condition is
expressed
(chapter Algebra) but with no rationale for its inclusion.

http://www.scribd.com/doc/18436832/Differential-geometry-reconstructed-a-unified-systematic-framework

[GrafZahl]

Hi fl,

My browser crashes when I try to access this document. My guess is
that all that the zero-ring condition is to express is that the zero
ring is not a field. For in a ring with unity R, all of the following
are equivalent:

* R is the zero ring
* One equals Zero
* Addition equals Multiplication (as functions from RxR -> R)

I have learned the following definition for a field:

R is a set together with functions +,·:RxR->R such that (R,+) is a
group whose neutral element we call 0, (R\{0},·) is an Abelian group
whose neutral element we call 1, and we have the distributive law x·(y
+z)=(x·y)+(x·z) for all x,y,z in R.

In this definition, the "R\{0}" effectively embeds the zero-ring
condition in the form "One does not equal Zero".

Best regards,
[[GrafZahl]]

[fl]

Hi GrafZahl,

> My browser crashes when I try to access this document.

Sure. Kennington should split up his book.

> For in a ring with unity R, all of the following are equivalent:

> * R is the zero ring
> * One equals Zero

Ah yes! I think it's the reason for the exclusion of the zero ring.
They want the one and the zero be different. In that case it would
be easy to replace the condition "is not the zero ring" by the
condition "the zero is different from the one".

The reason why I don't like the "is not the zero ring" condition
is that it doesn't look good in Metamath.

> R is a set together with functions +,·:RxR->R such that (R,+) is a
> group whose neutral element we call 0, (R\{0},·) is an Abelian group
> whose neutral element we call 1, and we have the distributive law x·(y
> +z)=(x·y)+(x·z) for all x,y,z in R.

I could also use a definition with groups that's true.

> In this definition, the "R\{0}" effectively embeds the zero-ring
> condition in the form "One does not equal Zero".

Thank you for this precision.

By the way a division ring looks pretty much like a field is there
a very good reason to keep both ?

--
FL

[fl]

> By the way a division ring looks pretty much like a field is there
> a very good reason to keep both ?

Definitely a field is a commutative division ring.

reference: http://en.wikipedia.org/wiki/Field_(mathematics)

So I think there's no need to define an extra structure. What we need
is a "switch" called comm2 that would coerce the second operation
to be commutative. For instance:

comm2 = { <. x , y. > | A. a e. ran y A. b e. ran y ( a y b ) = ( b y
a ) }

That way the definition of a field would be ( divRing i^i comm2 )

That switch could be used for all a range of rings of various flavors.

We could have the same switch for the structures with only one
operation
(group, monoid...)

comm1 = { x | A. a e. ran x A. b e. ran x a x b = b x a }