Filtered Ring Derived from Discrete Valuation Ring and Its Properties ()
1. Introduction
In commutative algebra, valuation ring and filtered ring are two most important structures (see [1] -[3] ). If
is a discrete valuation ring, then
has many properties that have many usages for example decidability of the theory of modules over commutative valuation domains (see [1] -[3] ), Rees valuations, and asymptotic primes of rational powers in Noetherian rings, and lattices (see [4] ). We know that filtered ring is also a most important structure since filtered ring is a base for graded ring especially associated graded ring, completion, and some results like on the Andreadakis Johnson filtration of the automorphism group of a free group (see [5] ) on the depth of the associated graded ring of a filtration (see [6] ). So, as important structures, the relation between these structures is useful for finding some new structure. In this article, we show that we can make a filtration with a valuation. Then we explain some new properties for it. On the other hand, we show this is a strongly filtered ring, then we explain some new properties for it.
2. Preliminaries
In this paper the ring
means a commutative ring with unit.
Definition 2.1 A subring
of a field
is called a valuation ring of
, if for every
,
, either
or
.
Definition 2.2 Let
be a totally ordered abelian group. A valuation
on
with values in
is a mapping
satisfying:
i)
;
ii)
.
Definition 2.3 Let
be field. A discrete valuation on
is a valuation
which is surjective.
Definition 2.4 A fractionary ideal of
is an
-submodule
of
such that
, for some
,
.
Definition 2.5 A fractionary ideal
is called invertible, if there exists another fractionary ideal
such that
.
Proposition 2.1 Let
be a local domain. Every non zero fractionary ideal of
is invertible if and only if
is DVR (see [3] ).
Theorem 2.1 Let
be a Noetherian local domain with unique maximal ideal
and
the quotient field of
. The following conditions are equivalent.
i)
is a discrete valuation ring;
ii)
is a principal ideal domain;
iii)
is principal;
iv)
is internally closed and every non-zero prime ideal of
is maximal;
v) Every non-zero ideal of
is power of
(see [3] ).
Definition 2.6 Let
be a ring together with a family
of subgroups of
if satisfying the following conditions:
i)
;
ii)
for all
;
iii)
for all
;
Then we say
has a filtration.
Definition 2.7 Let
be a ring together with a family
of subgroups of
if satisfying the following conditions:
i)
;
ii)
for all
;
iii)
for all
;
Then we say
has a strong filtration.
Example 2.1 Let
be an ideal of
, then
is a filtration that is called
adic filtration ring.
Definition 2.8 Let
be a filtered ring. A filtered
-module
is an
-module together with family
of subgroup
of satisfying:
1.
;
2.
for all
;
3.
for all
.
Then we say
has a filtration.
Definition 2.9 A map
is called a homomorphism of filtered modules, if: i)
is
-module an homomorphism and ii)
for all
.
Definition 2.10 A graded ring
is a ring, which can expressed as a direct sum of subgroup
i.e.
such that
for all ![](https://www.scirp.org/html/htmlimages\1-5300497x\85ae8c2c-1871-4b9e-80bd-3d7b2dfd4cca.png)
Definition 2.11 Let
be a graded ring. An
-module
is called a graded
-module, if
can be expressed as a direct sum of subgroups
i.e.
such that
for all
.
Definition 2.12 Let
and
be graded modules over a graded ring
. A map
is called homomorphism of graded modules if: i)
is
-module an homomorphism and ii)
for all
.
Definition 2.13 Let
be a filtered ring with filtration
. Let
, and
. Then
has a natural multiplication induced from
given
![](https://www.scirp.org/html/htmlimages\1-5300497x\6af73248-af7e-4278-a3bb-36098744fcf9.png)
where
. This makes
in to a graded ring. This ring is called the associated graded ring of
.
Definition 2.14 Let
be a filtered
-module over a filtered ring
with filtration
and
respectively. Let
, and
. Then
has a natural
-module structure given by
, where
.
3. Filtered Ring Derived from Discrete Valuation Ring and Its Properties
In this section we proved that, if
is a discrete valuation ring, then
is a filtered ring. And we prove some properties for
.
Let
be a field which
be a domain and a discrete valuation ring (DVR) for
. The map
is valuation of
.
Lemma 3.1 By above definition, the set
is an ideal of
.
Proof. (see [3] )
Theorem 3.1 If
is a discrete valuation ring with valuation
. Then
is a filtered ring with filtration defined by
![](https://www.scirp.org/html/htmlimages\1-5300497x\1aa05fcb-c804-457d-97cb-600cb4eed06c.png)
where
.
Proof. By definition of valuation ring, it is obvious that
. For the second condition for filtration ring we have
, So we have
.
For the third condition, we have for every
and
without losing generality. Since
and
are ideals of
so
![](https://www.scirp.org/html/htmlimages\1-5300497x\5a2ea398-4c4b-4f0f-b298-d956b326aec2.png)
is an ideal of
.
Now let
then
for
and
.
Thus
![](https://www.scirp.org/html/htmlimages\1-5300497x\a7155b1e-ae3e-4423-a6c5-35f9f33b3c4e.png)
Consequently we have
hence
. Therefore
is a filtered ring.
Proposition 3.1 Let
be a local domain. If every non-zero fractionary ideal of
invertible, then
is filtered ring.
Proof. By proposition 2.1
is DVR then by theorem 3.1
is filtered ring.
Proposition 3.2 Let
be a Noetherian local domain with unique maximal ideal
and
the quotient field of
. Then
is filtered ring if one of following conditions is held i)
is a principal ideal domain;
ii)
is principal;
iii)
is integrally closed and every non-zero prime ideal of
is maximal.
Proof. It follows from theorem (3.1) and theorem (2.1).
Definition 3.1 Let
be a ring, and let
be a totally ordered cancellative semigroup having identity
. A function
is a filtration if
,
and for all
i)
, and ii)
, then
is called a filtration.
For this filtration we have 1)
the set of ideals;
2)
;
3)
;
4)
.
Lemma 3.2 Let
be a filtration and let
. Then:
i)
;
ii)
;
iii)
;
iv) if
, then
and
.
Proof. See lemma 3.3 of [7] .
Proposition 3.3 If
be a discrete valuation ring, then there exists a totally ordered cancellative semigroup
, and
such that:
i)
;
ii)
;
iii)
;
iv) if
, then
, and
.
Proof. By theorem 3.1 there exists a filtration for
, then by lemma 2.1 we have the all above conditions.
Proposition 3.4 Let
be a filtered ring,
,
filtered
-modules, and
homomorphism of filtered
-modules. If the induced map
is injective, then
is injective provided
. (see [3] )
Corollary 3.1 Let
be a valuation ring,
,
filtered
-modules, and
homomorphism of filtered
-modules. If the induced map
is injective, then
is injective provided
.
Proposition 3.5 If
is a discrete valuation ring with valuation
, Then
is a strongly filtered ring with filtration defined by
![](https://www.scirp.org/html/htmlimages\1-5300497x\e9b1b345-fb35-4514-af2b-aaff2d9ef07a.png)
where
.
Proof. By theorem 3.1
is a filtered ring. Now we show
for all
. Since
so
![](https://www.scirp.org/html/htmlimages\1-5300497x\45b87854-fc08-4f02-92a4-112a2ce217af.png)
Consequently
, and
. Therefore
.
Proposition 3.6 Let
be a discrete valuation ring, and
. If
and
, then
is smallest prime ideal in
which contains
, and
is largest prime ideal in ![](https://www.scirp.org/html/htmlimages\1-5300497x\d1ae69f1-fcc6-4f1a-9a34-c07adada4922.png)
which does not contains
.
Proof. By proposition 3.5
is strongly filtered ring, then by proposition 4.2. of [7] -[9] we have If
and
, then
is smallest prime ideal in
such that contains
, and
is the largest prime ideal in
such that does not contains
.
Remark 3.1 Given a strong filtration
on a ring
, we say that a prime
in
is branched in
, if
cannot be written as union of prime ideals in
such that properly contained in
.
Corollary 3.2 Let
be a discrete valuation ring and
. Then a prime ideal
in
is branched in
if and only if
for some
.
Proof. By proposition 3.5
is strongly filtered ring, then by proposition 4.5. of [7] a prime ideal
in
is branched in
, if and only if,
for some
.
NOTES
*Corresponding author.