Title
On 2-STEGE EUCLIDEAN RING and LAURENT SERIES
Author(s)
AMANO, Kazuo
Citation
[岐阜大学教養部研究報告] vol.[22] p.[83]-[86]
Issue Date
1986
Rights
Version
岐阜大学教養部 (Dep. of Math., Fac. of Gen. Educ., Gifu
Univ.)
URL
http://repository.lib.gifu-u.ac.jp/handle/123456789/47593
※この資料の著作権は、各資料の著者・学協会・出版社等に帰属します。
83
On 2-STA GE E U CLI D E A N R I N G and L A U R E N T SE R I E S
Dedicated to Professor K . 0 hta on his 60th birthday
K azuo A M A N O
Dep. of M ath. , Fac. of Gen. E duc. , Gifu U niv
( Received Oct. 13, 1986)
1.
1ntrOduction
L et 7? be an integral dom ain.
L et yV : 7? → Z be a norm m ap satisfying N ( O) = 0, y ( α)
> O for ど
z≠ O, and yV( 油 ) = yV( α) 夙 b) .
T he dom ain 7? is called euclidean if for any α, みin
7? with ゐ≠ 0, there exist ・7, y in 尺 such that
α= φ 十y atld 八町 ) < X ( ろ) .
G.E .Cooke in [1] considered the foHowing possibility : for any ど
z, ゐin7? with ろ≠0, there
exist p, qげ , S in 7? such that
α= 油 十y, b゛ qy十s, and yV(s) く yV( ゐ).
A nd he caHed the domain 7? 2- stage euclidean・
ln[11, heprovedthattherearemany 2-stageeuclideandomainsthatarenoteuclidean
in the usual sense.
N ow let R χこ R [ [ χ ] ] [ χ - I ] be the ring of forma1 L aurent serieswith coefficient in R 、
T hen P. Samuel in [ 4] proved that if ム?χ is euclidean, 尺 is so. A lso F. Dress proved the
converse・in[21.
ln this paper, w e prove analogous results in 2- stage euclidean rings・
2. L aurent series.
Let R be an integral dom ain w ith a norm m ap yV : R → Z. L et titx = 人? [ [X ] ] [ X ¯1] be
the ring of form al L aurent series with coefficients in 尺 .
For
ブ=ΣらyE刄
χ
, 吋三
R,h
EZ,ah
≠
0
f≧ yz
we put j(ブ) = α。, and /( O) = 0. So n s a map from R χto 7? レ
L emma 1.
P roof.
Foy auy f , g 鏑 R 加ith g ≠ 0, チこ gn or, f = 即 十 z
ノバ ( φ 幸 0 mod t( 征
L et yz( resp. ん) be the lowest degree of / ( resp・ g ) . Se口 (y) = が (g ) 十 乙 where
q, yE R o T hen w e can w rite
84
K azuo A M A N O
△
zノ =
/
一
qX
一 憎
= り ひ 十higher degree terms.
lf γま O mod Z(g) , we stop since /0 ) 二 r幸 O mod /( g) ,
lf y≡ O mod /(g) , we similarly construct
り= z
ノーqlXh1-kg, (柘= order of め
and so on. lf the process stops after a finite number of steps, we obtain f = 卯 十 t・, 卵 )
牢 O m od /(が) .
Otherwise the infinite sum
zx= qχ 4 十 qIχhX- 十 … … 十 qnX ` ¯ノ
゛十 … …
makes sence, and w e obtain 仁 即 ・
3.
2-stage euclidean.
ぺ
Ve define a map N χI R → Z by
凡 ・(/ ) = X( べ/ )).
T hen w e obtain the following,
P roposition 2.
び R is 2 -stage 鰍 d ide皿 面 th yes加 d to N , R x is 2 -stage eMdidea11
With 粍S加d tO N χ= N ・ t、
Proof.
F or any ソ≒g in R χ with g≠ 0, by lemma l we have the following :
(1) 戸 即 ,
o「
(2) / = 即 十馬 ㈲ ) ま Omod /(g).
lt is obvious that the case ( 1) is good. A s for the case ( 2) , if X ( 府 ) ) < 餓 心 ) ) , then
we have N χ( r) く N χ(g) by definition, and this isgood. lf 入分 0 ) ) ≧X ( /(g)) , we have
心ノト が(g) + 7づ (g) = 肝十s, andⅣ(s) く M /(g)),
because 7? is 2-stage euclidean.
N ow if we set
z
/一1)X - 刄= が, (y
z= order of め,
then w e have / = 0 十 pX ゛ - ) g 十 が and /( が) = y、
lf w e set
g 一 qX ¯ T = z/″,
we have べど ) = s、 lf s≠0, we obtain
/ = 0 + 1)Xh
゛-k)g十が,イ = ・7X ゛ り 丿十び≒
and y v(g) = N (/(g)) > y (s卜 y x( の 。
lf s= 0, w(? have /( が卜 弘
On the other hand, since 躍 り ≧y ( j( g) ) , we get y ( ○ 二 1
On 2-ST A GE E UCLI DEA N RI N G and LA URE N T SERI ES
85
T herefore we have
丿= /(g)F l and 心・) = μ(g) 十C リ(g) = 0 十C 1丿(れ
T his is contradiction for XO ) ま O mod /(g) .
Proposition 3.
坏 R バ S 2- stage eudideαれ 面 th γes加 d to N χこ N ・t, 仇心 R iS蚕 SO
2-s叫le四di面四 面 th 粍s加d toN.
P roof.
L et α and ゐ be elements in R with ゐ≠ 0.
Since 7も , is 2- stage euclidean,
there eχist 1) , q,y,S m 7? x such that
α= 夕ろ十 y, b= 肛 十s, and N λS) < N 勁 ) ,
where
♪= j) kX
十 higher degree terms
q二 qlχ l
+ 1! igher degree terms
y二 y77xX ″十 higher degree terms
s二 恥 X
十 higher degree terms
T hen the following can happen :
(1) Let Nχ
( y) く y¥へ・( ろ). lf んく 0, wehave
ん= 附 and 夕丿 十y,7z= 0,
and hence
N式y) = X( ら) = yV(リ φ) ≧X( ろ) べ ¥y ろ).
T his is contradiction.
T herefore we get ん≧ 0 .
A nd we obtain
α= 瓦ろ十y0, N ( り卜 凡 脂 ) く Λ≒( ろ) = 躍 分
(2) Let yVじ
,y
( y) ≧N式b). lf /十刑く 0, weget /十m¯ ll and り ら 十亀= 0.
H ence
二
Nχ
(s) = 躍 如) = X (一々几 ) ≧躍 ら) ≧Λ≒( ゐ).
Since N χ( s) く y¥≒( ろ) , this is contradiction and hence we get /十 附 ≧0. T hen we can consider now possibility :
Case i) s。≠ 0 .
/
lf んく 0, we get み み十 戸 = 0.
0 n the other hand, since ろ= x
7げ m十 S0, w e have
躍 禎 = yV( み
-g几 ) こN(( 7十夕J ) ろ) ≧yV( ろ).
T his is contradiction, because yV( 貼) = N χ( s) く N χ( ろ) = yV( ろ) . H ence we have ん≧0. T hen
we obtain
と
z= 九 ろ十 γ
O, b= ㈲ り 十知, and yV( 知) く y O ) .
Case ii)・ s。= 0.
ln this case, w e distinguish now tw o subcases :
86
K azuo A M A N O
ノ
l n
乙
ダ
jj
lf ん≧Oj t is obviousthat α= 1)O
b十y0, b= ㈲秘十〇, and yV(O) < yV( ろ) .
lf ん< 0, we have ん= 辨 < O and 似 b十 ら = 0.
On the other hand, sincと ゐ= qlym, we have qd) h+ 1= O and hence (h, 1) kaTe units.
T hen w e can obtain :
卜 ( (hχl十……) ( 几万 十……) 十(稲X″
十……)
し
= のら 十(qlキlyk十のらu)X 十(ql午
2yk十眼Uyk+丿
十のり +2)X2十……十(亀X″十……)
T herefore we get the following equations :
qt+ j4 + qげ瓦+ j = 0
ql+ 2瓦 十 ・7, 十j 4 + 7十 qげ瓦や2= 0
● ●
●
φ
●
●
●
●
● ●
● ●
qa n瓦 十 ㈲ ゆ - j 4 + 7+ … … + qげ瓦十n+ 陥 = 0
Since qj s unit, we have
4 +7= φ¯lqGI瓦二ql¯lql±
1(一
九) 友
H ence we get yk十I ≡ O m od ゐ.
し
Similarly, we have
4 十2, ‥‥ , 4 十,2- j 三O mod &
T hen if ん十 万< 0, w e have 1) k nb十 八 十,7= O and hence yk+ 1≡ O m od ろ.
By above equations ( E ) , w e have 亀 ≡ O m od ゐ and hence y ( 如 ) ≧ y ( ゐ) . T his 沁 contradiction to N ( 如) < X ( 分
T herefore w e get ん十 刀≧ 0・
N ow, if ん十 刀> 0, there eχist an integer /xsuch that ん十 刀= 0. T hen since 弘 .14 十 …
… + 9几 n = 0, we have 勾 = a n ≡ O m od & .
H ence w e obtain a = 九φ 十 勾 = q b.
lf ん+ 77= 0, the equation (E ) we have
q卜n几十 …・ 十哨ら+,2= qanyk十 …・ 十の○ 一九ろ) 十亀= 0. 1
T hen w e obtain
aこ 1)ob十の-j(-みゅ4 - ‥‥一
亀) = がろ十び/ (一亀), and yV(g/ (一亀)) = yV(如) < X( ろ).
R eferenCes
[1] G.E.Cooke, A weakening of theeuclideanpropertyforintegral domainsandapplicationstoalgebraic
number theory.I , J.Reine A ngew. M ath.282 (1976), 133- 156
[2] F.Dress, Stathmeseudidiensetseriesformelles, ActaArith.,χIχ (1971), 261-265
[3] H.W.Lenstra,紅 , Lectures onEuclideanrings, Bielefeld 1974
閻
P.Samue1, AboutEuclideanRings, J.Algebra 19 (1971)√282-301

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