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L For a graded ring R, a graded left R-module M is a left R-module with a grading M = M γ for , γM γ R 0 where the all abelian and abelian such that γ · M δ M γ is containing groups group are a + δ ,

all γ

# 0. Then,

## M

and Mh are defined analogously to

R

and Rh. We say that M is a graded

free R-module if it has a base as a free R-module consisting of homogeneous elements.

=

Lγ

E γ

division

ring

if

is a torsion-free abelian group and every

non-zero homogeneous element of E has a multiplicative inverse. Note that the grade set

## E

is actually

a γ

g r o u p . A l s o , E 0 i s a d i v i s i o E. The requirement that n

r i n g , a n d E γ i s a 1 - d i m e n s i o n a l l e f t a n d r i g h t E be torsion-free is made because we are interested 0

vector space for every in graded division rings

arising

from

valuations

on

division

rings,

and

all

the

groups

appearing

there

are

torsion-free.

Recall

that every example,

torsion-free embeds in

abelian group

admits total orderings compatible with the group structure. (For

# By using any total ordering on

E, it is easy to see that E

has no zero divisors and that

E, the multiplicative group of units of E, coincides with Eh \ {0} (cf.

[HwW2], p. 78).

# Furthermore, the

degree map

deg: E

E

(2.1)

i s a g r o u p h o m o m o r p h i s m w i t h k e r n e l E 0 .

By an easy adaptation of the ungraded arguments, one can see that every graded module M over a graded division ring E is graded free, and every two homogenous bases have the same cardinality. We thus call M a graded vector space over E and write dimE(M) for the rank of M as a graded free E-module. Let S E be a graded subring which is also a graded division ring. Then, we can view E as a graded left S-vector space, and we write [E : S] for dimS(E). It is easy to check the “Fundamental Equality,”

[ E : S ] = [ E 0 : S 0 ] | E : S | ,

w h e r e [ E 0 : S 0 ] i s t h e d i m e n s i o n o f E 0 a s a l e f t v e c t o r s p a c e o v e r t h e d i v i s i o n r i n g S 0 the index in the group E of its subgroup S. a n d |

E

:

(2.2) S| denotes

A graded eld T is a commutative graded division ring. Such a T is an integral domain, so it has a quotient field, which we denote q(T ). It is known, see [HwW1], Cor. 1.3, that T is integrally closed in q(T ). An extensive theory of graded algebraic extensions of graded fields has been developed in [HwW1]. For a graded field T , we can define a grading on the polynomial ring T [x] as follows: Let âˆ† be a totally ordered

abelian group with T [x] = T

âˆ†, and fix θ âˆ†. We have

L

T [x]γ ,

where

T [x]γ

= {P

aixi | ai T h, deg(ai) + = γ}.

(2.3)

γâˆ†

makes

# T [x]

a

ring,

which

we

denote

T [x]θ .

Note

that

T [x]

=

T

+

hθi.

A

homogeneous

polynomial in T [x]θ is said to be θ-homogenizable. If E is a graded division algebra with center T , and a Eh is homogeneous of degree θ, then the evaluation homomorphism î€„a : T [x]θ T [a] given by f 7→ f(a) is a graded ring homomorphism. Assuming [T [a] : T ] < , we have ker(î€„a) is a principal ideal of T [x] whose unique monic generator ha is called the minimal polynomial of a over T . It is known, see [HwW1],

Prop. 2.2, that if deg(a) = θ, then ha is θ-homogenizable. If E is a graded division ring, then its center Z(E) is clearly a graded field.

rings

considered

in

this

paper

will

always

be

assumed

nite-dimensional

over

their

centers.

The

finite-

dimensionality assures that E has q(E) = E T q(T ) where T = Z(E). E is defined by ind(E)2 = [E : T ]. If

a quotient division ring q(E) obtained by central localization, i.e., Clearly, Z(q(E)) = q(T ) and ind(E) = ind(q(E)), where the index of S is a graded field which is a graded subring of Z(E) and [E : S] < ,

then E is said to be a graded division algebra over S. A graded division algebra E with center T is said to be unrami ed if

E

=

T . From (2.2), it follows

t h e n t h a t [ E : S ] = [ E 0 : T 0 ] . A t t h e o t h e r e x t r e m e , E i s s a i d t o b e t o t a l l y r a m i e d i f E 0 = T 0 . I n a c a s e i n t h e m i d d l e , E i s s a i d t o b e s e m i r a m i e d i f E 0 i s a fi e l d a n d [ E 0 : T 0 ] = | E : T | = i n d ( E ) . T h e s e

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