Title
ユークリッド互除法に関する一注意
Author(s)
尼野, 一夫
Citation
[岐阜大学教養部研究報告] vol.[17] p.[55]-[56]
Issue Date
1981
Rights
Version
岐阜大学教養部 (Dept. of Math., Fac. of Gen. Educ., Gifu
Univ.)
URL
http://repository.lib.gifu-u.ac.jp/handle/123456789/47505
※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。
55
A
remark on E uclidean algorithm
Dedicated to P rofessor S. T omatsu on his 60th birthday
K azuo A M A N O
Dept. of M ath. , Rlc. of Gen. E duc. , Gifu U niv
(Received Oct. 5, 1981)
I.
l ntroduction.
tativity is required.
L et 沢 be a zero-divjsor-free ring.
Neither unity element nor commu-
T he customary definition of a 1Qft E uclidean algorithm in 尺 is a map 9)
of R into the set of nonnegative integers such that Q( 必 ) ≧ 9)( α), 衣 的
given (1, heR , b≠ 0, there exist g and r in R satisfying
α二 φ 十 r
for 必 中 O and that
w ith 祠 r) く 似 征
W e say
also that R is a left E uclidern ring with respect to this algorithm 9 .
0 n account of the duality the meaning of a right E uclidean ring is evident.
lf furthernlore
a ring is simultaneously left and right E uclidean, than we call it a E uclidean ring.
T he main interest of a left ( resp. righO E uclidean ring 瓦 is that it is a principal left (resp.
right) ideal ring with unity element and that, if the ring 尺 is, moreover, Euclidean, its ideals
are the principal ideals generated by the elements of the nornlaliser of 尺 〔2, T heorem 191〕 .
ln the proof, the inequality 9 ( 必 ) ≧ 9 ぐb) for 砧 中 0 has an important
role.
But, for a
零
commutative ring w ith unity elew ent,
P . Samuel 〔3〕 and T h. M otzkin 〔 1〕 noticed that the
inequality is unnecessary and that any we11-ordered set can replace the set of nonnegative integers as the r ange of 7 .
T herefore the hypothesis that an algorithm h卵
the inequality and
that it has the set of nonnegative integers as its range does not seem to be essentia1.
II .
GeneralizatiOn.
L et q be a left Euclidean algorithm as before but where we allow
Q to take a w e11- ordered set as the range; w e do not require the inequality 似 必 ) ≧ 以 α) , 以 /))
70r 必 中0.
T hen we can obtain the followings in the quite same methods as L . Redei 〔2〕
and P . Samue1 〔3〕 .
j°
.
凡 7` 昨
凡
け 0, u)e haue (p( ろ) > Q( O) , SO th t 収 O) 18 the 8・ (1Lte8t d ement of 9 ( 栢
〔3,
Prop. 1〕.
2°
.
R h(18 a uni砂 d ement 〔2レ pp. 325-326〕 .
Combining 1° with 2°
, we have
J°
.
A n d ement bE R 8uch 伍 (1t 収 b) 18 the 8m (1ne8t d ement of 9 ( 尺) 一
瓦 〔3, Prop. 2〕.
f.
祠 O) 18 a unit 伍
犬
Eりe巧 L示 ide(LL of the L示 Eucade皿 71昭 is a princi郊 に 示 ide(tL 〔3, Prop.
3〕.
A left Euclidean algorithm does not necessarily satisfy the ineqUality, but we can construct
the following in the same way as 〔3〕 .
56
K azuo A mano
ぎ.
The m叩 (pi, d岨 ned by (球 O) 二 涙O) ゛ ld gi( ) 二igL yQ) 釦T a ≠Oj 8 (1 td t Eudidean
d goTithm such that 9) i( 加) ≧ 卯 ( c) 釦r bc≠ O (1nd 8uch thc
tt g i( 加) > 卯 ( c) iJ b 18 not (t 皿 琵 〔3,
Prop. 4〕.
ln fact 炉i is well defined since the range of 9) is well ordered. ・T he first prope17ty follows
from the definition of 卯 .
have 卿(昂 =
L et us consider ふ≠ O in 尺 and a in 凡 By definitition olf 卯
涙砧) for a suitable c in 凡
w rite a= 叩 ろ十 r w ith 涙 r) < 衣 雨 .
left E uclidean algorithm.
Since p is a left Euclidean algorithm, we can
T herefore w e have り ( r) ≦ 削 節 ) =
り (征
H ence 卯 is a
F or proving the second property, we can write c= 昴 c+ r with 卯
(r) < り (加), because qi is a left Euclidean algorithm.
卯(r) = 7)i ((j - φ) c) ≧卿 (c).
we
By the first property, we have
Therefor we obtain 卯(加) > Q)i( r) ≧卿(cレ
On account of the duality the properties j °一 ダ are satisfied in right Euclidean ring.・
W e noxv asunle that R is an E uclidean.
principal left、and
a principal right ideal.
gorithm 卿 as in the property ダ .
Redei s book 〔2〕 .
(ミ
r.
By the property J°
, every ideal of 沢 is both a
F urthermore the ring 沢 has a left E uclidean a1-
H ence the algorithm qi satisfies the same situation as
T herefore we have the following.
Eりe巧 1ded of 皿 Eud idean 7謎g is 伍e F inciで
pd 謎ed genemted by 脹e normd iser (汀
凡 〔2, Theorem 191〕.
III.
E χample.
L et 尺 be a local field with respect to a normalized discrete valuation ひ
and let 工) be a division algebra over 瓦
W e consider a map g of 7) into the set of inte.
gers given by 以J) = む(爪J)) for xe D, wherey
Vis thereducednorm. Then it is wel卜kown
that 沢= 佃∈川 副J) ≧O} is a uniquemaximla10rder inDandthat every ネideal j is generated by an element a E/1 such that 切 ( a) is the smallest element of g ( ノ1) .
W e now consider a map 匹 of 沢 into the set of nonnegative integers, defined by 回 (O) = 0
and匹 ( y) = 1+ g (J) for any J≠O in R、 Then切i・ 18 (1n Eudidean dgoTitkm in R 8at18bing
め(晶) ≧n (α
), 回 0 ) for 晶 ≠0,
FurtheTmoTe for (1ny Eud謎e(1n dgoTithm 9) on R, 扨e h(1ue
衣J) ≧肪(J) foT (1ny x≠Oin R, henceuJ118 the8maUe8t dgori峨m. ln fact the first assertion
is trivia1.
F or proving the second property, w e may. replace 9) by 卿
卯(J) for any X in 凡
as in ダ , since 涙 功 ≧
First we havり 卯(司 ≧ぬGz) = 1 for any unit u.
Second we assume
that we have already proved our assertion for smaller values of 卯 (功.
Since we can write
x¯ u了
r゛(べwhere 77・is a prime element in 凡 we have
卯(J)2 卯(♂(勺 > 叫 が ( )¯1) ≧凹 ( π゛ ( )¯1) = g(ヱ
), hence り臨) ≧回(乱
R eferences
〔1〕
Th. M otzkin, 0 n the E uclidean algorithm, Bu11. A mer. M ath Soc. 55 ( 1949) , 1142- 1146.
[:2]
L. Redei, Algebra, Vol. 1, Hungary ( 1967) .
〔3〕
P、Samue1, A bout E uclidean rings, J. A lgebra 19 ( 1971) , 282- 30 1.

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