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DEFINITION 4.1
A set

of subsets of a set

is called
a
filter on

if the following conditions are
satisfied.
-
.
-
.
DEFINITION 4.2
A maximal filter on a set

is called an
ultra filter on

.
Those readers who are not familiar with the arguments are invited to read
for example [15] or the book of Bourbaki [2].
LEMMA 4.3
Let
be a filter on a set
The following statements are equivalent.
-
is an ultrafilter. That means, a maximal filter.
- for any subset
, we have either
or
DEFINITION 4.4
A principal filter on a set

is an ultra filter of the form

where

is an element of

.
A ultrafilter which is not principal filter is called non-principal.
An ultrafilter
on a set
may
be identified with a point
of Stone-Cech compactification of (
with discrete topology).
A non principal ultrafilter is identified with a
boundary point.
DEFINITION 4.6
Let

be a number field with the ring
of integers

.
Let

be a non-principal ultrafilter on the set

of all primes of

of height 1.
Let
be an ideal of
defined as follows:
Then we define a ring

as follows:
We denote by

the canonical projection from

to

.
PROOF..
Indeed, let

be a non zero element in

.
Let

.
Then for any

,
intersection

is non empty.
Maximality of

now implies that

itself is a member of

.
The inverse

of

in

is given by the following formula.
If

in

for a positive integer

,
then there exists

such that

. On the other hand,
as we have mentioned in Lemma
4.5 above,
being a member of a non-principal filter

,

cannot be a finite set.
This is a contradiction,
since non-zero member

in

has only finite ``zeros'' on
the ``arithmetic curve''

.
Thus the characteristic of

is zero.
The definition above is partly inspired by works of
Kirchberg (See [12] for example.)
We would like to give a
little explanation on
. We regard it as a kind of `limit'.
If we are given a member
of
and
we have an element, say,
of
for each primes
, then, by assigning
arbitrary element to `exceptional' primes (that means, primes which
are not in
), we may interpolate
and
consider
The element ('limit') does not actually depend on
the interpolation. Thus we may refer to the element without specifying the
interpolation. In particular, this applies to the case where we have
for almost all primes
.
The same type of argument applies for polynomials. We summarize this in the
following Lemma.
LEMMA 4.8
Suppose we have a co-finite subset
of
and a collection
of polynomials. Assume we have a bound
for the degrees of the polynomials.
That means,
Then we may define the `limit'
by taking `limit' of each of the coefficients.
The same arguments also applies for polynomial maps.
For any non-principal ultra filter
on
(prime numbers)
,
We may consider the following ring.

$U$ �� $0$
It turns out that,
Thus we conclude that
PROPOSITION 4.10
As an abstract field,
Next: Elementary category theory
Up: Topics in Non commutative
Previous: locally free sheaves of
2007-12-11