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arXiv:1209.4172v1 [math.AC] 19 Sep 2012

Quasi-Valuations Extending a Valuation

Shai Sarussi∗

Department of Mathematics

Bar Ilan University

Ramat-Gan 52900, Israel

Abstract. Suppose F is a field with valuation v and valuation ring Ov, E is a

finite field extension and w is a quasi-valuation on E extending v. We study quasi-

valuations on E that extend v; in particular, their corresponding rings and their

prime spectrums. We prove that these ring extensions satisfy INC (incomparability),

LO (lying over), and GD (going down) over Ov; in particular, they have the same

Krull Dimension. We also prove that every such quasi-valuation is dominated by

some valuation extending v.

Under the assumption that the value monoid of the quasi-valuation is a group we

prove that these ring extensions satisfy GU (going up) over Ov, and a bound on the

size of the prime spectrum is given. In addition, a 1:1 correspondence is obtained

between exponential quasi-valuations and integrally closed quasi-valuation rings.

Given R, an algebra over Ov, we construct a quasi-valuation on R; we also con-

struct a quasi-valuation on R ⊗OvF which helps us prove our main Theorem. The

main Theorem states that if R ⊆ E satisfies R∩F = Ovand E is the field of fractions

of R, then R and v induce a quasi-valuation w on E such that R = Owand w extends

v; thus R satisfies the properties of a quasi-valuation ring.

§0 Introduction

Recall that a valuation on a field F is a function v : F → Γ∪ {∞}, where Γ is a

totally ordered abelian group and v satisfies the following conditions:

(A1) v(0) = ∞;

(A1’) v(x) ?= ∞ for every 0 ?= x ∈ F;

(A2) v(xy) = v(x) + v(y) for all x,y ∈ F;

(A3) v(x + y) ≥ min{v(x),v(y)} for all x,y ∈ F.

Valuation theory has long been a key tool in commutative algebra, with applica-

tions in number theory and algebraic geometry. It has become a useful tool in the

study of division algebras, and used in the construction of various counterexamples

such as Amitsur’s construction of noncrossed products division algebras. See [Wad]

for a comprehensive survey.

Generalizations of the notion of valuation have been made throughout the last

few decades. Knebusch and Zhang (cf. [KZ]) have studied valuations in the sense of

Bourbaki [Bo, section 3]. Thus they were able, omitting (A1’), to study valuations

on any commutative ring rather than just on an integral domain. They define the

notion of Manis valuation (the valuation is onto the value group) and show that an

R-Pr¨ ufer ring is related to Manis valuations in much the same way that a Pr¨ ufer

domain is related to valuations of its quotient field.

∗This paper is part of the author’s forthcoming doctoral dissertation.

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A monoid M is called a totally ordered monoid if it has a total ordering ≤, for

which a ≤ b implies a + c ≤ b + c for every c ∈ M. When we write b > a we mean

b ≥ a and b ?= a.

In this paper we study quasi-valuations, which are generalizations of valuations.

A quasi-valuation on a ring R is a function w : R → M ∪ {∞}, where M is a

totally ordered abelian monoid, to which we adjoin an element ∞ greater than all

elements of M, and w satisfies the following properties:

(B1) w(0) = ∞;

(B2) w(xy) ≥ w(x) + w(y) for all x,y ∈ R;

(B3) w(x + y) ≥ min{w(x),w(y)} for all x,y ∈ R.

In the literature this is called a pseudo-valuation when the target M is a totally

ordered abelian group, usually taken to be (R,+). As examples one can mention

Cohn (cf. [Co]) who gave necessary and sufficient conditions for a non-discrete topo-

logical field to have its topology induced by a pseudo-valuation; Huckaba (cf. [Hu])

has given necessary and sufficient conditions for a pseudo-valuation to be extended

to an overring, and Mahdavi-Hezavehi has obtained ”matrix valuations” from ma-

trix pseudo-valuations ([cf. MH]). We use the terminology quasi-valuation to stress

that the target monoid need not be a group. Moreover, our study concentrates on

quasi-valuations extending a given valuation on a field.

The minimum of a finite number of valuations with the same value group is a

quasi-valuation. For example, the n-adic quasi-valuation on Q (for any positive

n ∈ Z) already has been studied in [Ste]. (Stein calls it the n-adic valuation.) It

is defined as follows: for any 0 ?=

a,b ∈ Z, with b positive, such that

Define wn(c

A quasi-valuation is a much more flexible tool than a valuation; for example,

quasi-valuations exist on rings on which valuations cannot exist.

Three main classes of rings were suggested throughout the years as the non-

commutative version of a valuation ring. These three types are invariant valuation

rings, total valuation rings, and Dubrovin valuation rings. They are interconnected

by the following diagram:

c

d∈ Q there exists a unique e ∈ Z and integers

c

d= ne a

bwith n ∤ a, (n,b) = 1 and (a,b) = 1.

d) = e and wn(0) = ∞.

{invariant valuation rings}⊂{total valuation rings}⊂{Dubrovin valuation rings}.

Morandi (cf. [Mor]) has studied Dubrovin valuation rings and their ideals. It

turned out that unlike a valuation on a field, the value group of a Dubrovin valuation

ring B does not classify the ideals in general but does so when B is integral over

its center.

At the outset and in section 1 one sees that there are an enormous amount of

quasi-valuations, even on Z, so in order to obtain a workable theory one needs

further assumptions. Morandi (cf. [Mor]) defines a value function which is a quasi-

valuation satisfying a few more conditions. Given an integral Dubrovin valuation

ring B of a central simple algebra S, Morandi shows that there is a value function

w on S with B as its value ring (the value ring of w is defined as the set of all

x ∈ S such that w(x) ≥ 0). Morandi also proves the converse, that if w is a value

function on S, then the value ring is an integral Dubrovin valuation ring.

The use of value functions allow a number of results about invariant valuation

rings to be extended to Dubrovin valuation rings. Also, their use has led to simpler

and more natural proofs of a number of results on invariant valuation rings.

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Tignol and Wadsworth (cf. [TW]) have developed a powerful theory which uti-

lizes filtrations. They consider the notion of gauges, which are the surmultiplicative

value functions for which the associated graded algebra is semisimple, and which

also satisfy a defectlessness condition. The gauges are defined on finite dimensional

semisimple algebras over valued fields with arbitrary value groups. Tignol and

Wadsworth also show the relation between their value functions and Morandi’s.

Quasi-valuationsgeneralize both value functions and gauges (the axioms of quasi-

valuations are contained in the axioms of value functions and gauges). Although

we do not obtain Tignol and Wadsworth’s decisive results concerning the Brauer

group, we do get a workable theory, in which we are able to answer questions regard-

ing the structure of rings using quasi-valuation theory. One difference with value

functions, for example, is that on a field, Morandi’s construction of a value function

is automatically a valuation, while that is not the case for quasi-valuations. Gauges

on a field reduce to exponential quasi-valuations, a special case of quasi-valuations.

We believe that, even over a field, quasi-valuations are a natural generalization of

valuations, and enrich valuation theory even for commutative algebras.

Whereas for a valuation on a field M is automatically a group since

v(x−1)=

−v(x),

w(x−1) + w(x) ≤ w(1)).

case w : R → M ∪ {∞} is a quasi-valuation for M a group, we say that w is

group-valued (rather than calling it a pseudo-valuation); we do not require w to be

surjective.

Note that it does not follow from the axioms of a quasi-valuation that w(1) is

necessarily 0. We shall see examples in which w(1) ?= 0.

We list here some of the common symbols we use for v a valuation on a field F

and w a quasi-valuation on a ring R (usually we write E instead of R):

Ov= {x ∈ F | v(x) ≥ 0}; the valuation ring.

Iv= {x ∈ F | v(x) > 0}; the valuation ideal.

Ow= {x ∈ R | w(x) ≥ 0}; the quasi-valuation ring.

Iw= {x ∈ R | w(x) > 0}; the quasi-valuation ideal.

Jw; the Jacobson radical of Ow.

Γv; the value group of the valuation v.

Mw; the value monoid of the quasi-valuation w, i.e., the submonoid of M gen-

erated by w(R \ {0}).

Note that w is group-valued iff Mw is cancellative, since any ordered abelian

cancellative monoid has a group of fractions.

Here is a brief overview of this paper. In section 1 we present some general

properties of quasi-valuations on rings as well as some basic examples. Thereafter

we work under the assumption that F denotes a field with a valuation v, E/F

usually is a finite field extension, and w is a quasi-valuation on E such that w|F = v.

In section 2 we discuss some of the basic results regarding quasi-valuations and

their corresponding rings; most of the results in this section are valid in the more

general case where E is a finite dimensional F−algebra. In section 3 we prove

that a quasi-valuation ring satisfies INC and LO over Ov; in fact, LO is valid in

the case where E is a finite dimensional F−algebra. In section 4 we introduce a

notion called PIM (positive isolated monoid); the PIMs enable us to prove that a

quasi-valuation ring satisfies GD over Ov. We also describe a connection between

the closure of a PIM and the prime ideals of Ow. In sections 5 and 6 we assume

that w(E \ {0}) is torsion over Γv, the value group of the valuation. In section

the situation is different with quasi-valuations (since

Thus it is more natural to map w to a monoid. In

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5 we generalize some properties of valuation rings and we discuss the set of all

expansions of a quasi-valuation ring (and one of his maximal ideals). We show that

a quasi-valuation ring satisfies GU over Ov. We also construct a quasi-valuation

that arises naturally from w; this quasi-valuation is one of the key steps to give

a bound on the size of the prime spectrum of the quasi-valuation ring. In section

6 we prove that any valuation whose valuation ring contains Ow dominates w.

We also show that Ow satisfies the height formula (since we have the property

GU). In section 7 we discuss exponential quasi-valuations; we do not assume that

w(E \ {0}) is torsion over Γv. Instead, we assume the weaker hypothesis that the

value monoid is weakly cancellative. We prove that an exponential quasi-valuation

must be of the form w = min{u1,...,uk} for valuations uion E extending v. We

obtain a 1:1 correspondence between {exponential quasi-valuations extending v}

and {integrally closed quasi-valuation rings}. We also deduce that the number of

exponential quasi-valuations is bounded by 2[E:F]− 1. In section 8 we construct

a total ordering on a suitable amalgamation of Mw and Γdiv that allows us to

compare elements of Γdiv with elements of Mw. Then we show that there exists

a valuation u whose valuation ring contains Ow and u dominates w. In section 9

we review some of the notions of cuts and we present the construction of the cut

monoid (of a totally ordered abelian group) which is an N-strictly ordered abelian

monoid. Then we introduce the filter quasi-valuations, which are quasi-valuations

that can be defined on any Ov−algebra. The filter quasi-valuation is induced by

an Ov−algebra and the valuation v; its values lie inside the cut monoid of Γv. This

gives us our main theorem, which enables us to apply the methods developed in the

previous sections, as indicated below in Theorem 9.37. In section 10 we show that

filter quasi-valuations and their cut monoids satisfy some properties which general

quasi-valuations and their monoids do not necessarily share. This enables us to

prove a stronger version of Theorem 8.15 when dealing with filter quasi-valuations.

We also show that the filter quasi-valuation construction respects localization at

prime ideals of Ov. Finally, we present the minimality of the filter quasi-valuation

with respect to a natural partial order.

§1 General Quasi-Valuations and Examples

In this section we present some of the basic definitions and properties regarding

quasi-valuation on rings. We also present some examples of quasi-valuations. In

particular, we give examples of quasi-valuations on integral domains which cannot

be extended to their fields of fractions.

Remark 1.1. If the monoid M is cancellative then w(1) ≤ 0.

Proof. w(1) = w(12) ≥ w(1) + w(1).

Here is a trivial example of a quasi-valuation on Z.

Example 1.2. w(0) = ∞ and w(z) = −1 for all z ?= 0.

Lemma 1.3. Let w be a quasi-valuation on a ring R and suppose that w(−1) = 0.

Then w(a) = w(−a) for any a ∈ R.

Proof. Let a ∈ R; then

w(−a) = w(−1 · a) ≥ w(−1) + w(a) = w(a).

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By symmetry w(a) = w(−(−a)) ≥ w(−a).

The following lemma generalizes a well known lemma from valuation theory,

with the same proof. We prove it here for the reader’s convenience.

Lemma 1.4. Let w be a quasi-valuation on a ring R and suppose that w(−1) = 0.

Let a,b ∈ R;

(i) If w(a) ?= w(b) then w(a + b) = min{w(a),w(b)}.

(ii) If w(a + b) > w(a) then w(b) = w(a).

Proof. We prove the first statement; the second follows easily. Assume w(a) < w(b);

then w(a + b) ≥ w(a). On the other hand,

w(a) = w(a + b − b) ≥ min{w(a + b),w(b)} = w(a + b).

Q.E.D.

In order to be able to use techniques from valuation theory, we define the fol-

lowing special elements with respect to a quasi-valuation.

Definition 1.5. Let w be a quasi-valuation on a commutative ring R. An element

c ∈ R is called stable with respect to w if

w(cx) = w(c) + w(x)

for every x ∈ R.

Lemma 1.6. Let w be a quasi-valuation on a commutative ring R such that w(1) =

0. Let c be an invertible element of R. Then c is stable iff w(c) = −w(c−1).

Proof. (⇒) c is stable and invertible; thus

0 = w(1) = w(cc−1) = w(c) + w(c−1),

i.e., w(c) = −w(c−1).

(⇐) Let x ∈ R; we have

w(x) = w(c−1cx) ≥ w(c−1) + w(cx)

≥ w(c−1) + w(c) + w(x) = w(x).

Hence equality holds, and since w(c−1) = −w(c) is invertible, we have

w(cx) = w(c) + w(x).

Q.E.D.

Definition 1.7. An exponential quasi-valuation w on a ring R is a quasi-valuation

such that for every x ∈ R,

w(xn) = nw(x),(1)

∀n ∈ N.

We note that any exponential quasi-valuation w with Mw cancellative satisfies

w(1) = w(−1) = 0.

Usually we need only check the following special case of (1):