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 
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
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
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
is an ideal of
.
Now let 
then 
for 
and
.
Thus
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
where
.
Proof. By theorem 3.1 
is a filtered ring. Now we show 
for all
. Since 
so
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 
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.