Title
ユークリッド環についての注意
Author(s)
尼野, 一夫
Citation
[岐阜大学教養部研究報告] vol.[20] p.[13]-[15]
Issue Date
1984
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/123456789/47553
※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。
13
A N OT E ON
E U CL I D E A N
RI N G
Dedicated to Professor K . Y amamoto on his 60th birthday
K azuo A M A N O
Dep.
of M ath. , F ac.
of Gen.
(Received Oct.
1
12,
E duc. , Gifu U niv
1984)
l ntroduction.
A ring tit is called a Euclidean ring if there is a map t of tit into a wen-ordered set I F such that,
given a, bE R , b≠ 0, there exist q and y in 尺 such that ど
z= 匈 + y, and r = O or
7(y) く 7( ろ).
T he map g is called a E uclidean algorithm in R 。
N ow, let ヵ be a rational prime, and let Qg be the p-adic completion of the field
Q of all rational numbers.
W e call the ター
adic limit of a sequence of rational inte-
gers ター
adic integer, and we denote by Z7, the ring consisting of the ター
adic integers.
T hen, it is w e11-known [5, Prop. 5] that Zr, is a E uclidean domain with a E uclidean
algorithm 7 (x) = 1 十 zノ(x) , where z・ denotes the normalized ヵ-adic valuation in Q.
0 n the other hand, 乙 is the projective limit of the projed ive system { Z/ y Z
O= 1, 2, ‥ 。}, 昿 : Z/芦 Z-
Z/ダZ} 。
Zp ≧ 11m Z/ y Z,
where Z denotes the ring of rational integers.
M oreover Z/ y Z is a E uclidean ring
with an algorithm t which is defined by :
7(x) = min{ ¦列 レ E x十ダ Z, x十ダ Zと Z/ダ Z}.
By m ean of above argument, we can establish the following problem in general ;
£d { 7?μ77こ 1, 2,‥・}, 鵬 : Rm→
Rn(ring hQm.) } h a 隙ojec面esyste琲 of 沁lgsRn.
仔辿 侑e Rn aR EMd 面四 百紹s, 侑四丿 s 侑e y ojec面 e l泌
= lim Rn a EMdidea11
riug P
ln the present paper,
of above problem .
we will give an another example which shows a validity
N amely,
K azuo A m ano
14
T HEOREM. L d g a 叩tio皿 口 ㎡ egey 切hich is Mot 岡皿 口 o 犯 tiom l j) yime.
T heu, 山e
加伽 d紬八 i面 t Z90f the 鈴融 d面 sが em { Z/が Z印= 1, 2, ‥ ・} , πら: Zソが Z一
Z/が Z} .
Zg= 迦 Z/がZ,
is α E Mclideau y鏑g sud
tk l il is ilot alx integral domαiu ayld j oγally m卯 l- : Zg→
(the set of Mou一
麗即 t加e 鏑t昭eys) c
皿 肴ot be a EMdide皿 ri昭 .
N
The血dor gro呻 Z/が Z
is ㎡so a E ud ideall riug Joy .ll = 1, 2, ‥ 二
2.
P roof of theorem.
lt is trivial that Z/ が Z is a E uclidean ring with an algorithm φ defined by :
φ(x) = 肩肘 に㈱ yE x十がZ, ズ十が ZE Z/が Z} .
Let ヵ1, 1)2, ‥ ・ , 九 be distind primes, and let
g= 斑1‥. 夕v ahd が= 耳1‥. 1)に
where the eχponent y1, ‥ . , ら , 汀 , ‥ . , りに are positive integersバ T hen we have g -
adic (resp. が-adic) completion Qg (resp. Qy) of Q.
are not essentially distinct [31.
The ring qg and the ring qg,
べ
Ve may therefore assume that g= 九 ‥ . 九 .
T hen,
by Chinese remainder theorem, w e have
Z/が Z ⊇ Z/班Ze ‥ . eZ/詞Z
for 刀= 1, 2/ ‥ .
Passing to the projective lim it, w e obtain
Zg≧Zか (D ‥ . (DZ八 ・
P、 Samuel showed that a product of a finite number of E uclidean ring is a Euclid-
ean ring [5, Prop. 6].
Since the ring of the 良 一
adic integers Zg is a Euclidean do-
main, w む get that Zg is a E uclidean ring.
0 f couse, Zg is not domain since it has
zero-divisors.
0 n the other hand, C. R . F letcher showed that, if a ring 尺 is a Euclidean and
7? = 7? 1(D . ‥ (Dtita where カ > 1, then 7? f is a bounded E uclidean ring for i= 1, 2/ ‥ ,
k, where a E uclidean ring 7? is said to be bounded if it possesses an algorithm z・: £
一
N w ith property that r ( y) is bnunded for all non-zero yE R [2, Prop. 51.
T here-
fore, if w e assume that Zg is a E uclidean ring with an algorithm 9 : Zg→
N , we
have that Zゑ is a bounded E uclidean domain.
A bounded E uclidean domain is a
field [2, Prop. 3]. But Zg is not a field. This is unreasonable.
3.
R emarks.
(1)
The customary range of a Euclidean algorithm istheset N of the non-neg-
ative integers.
algorithm .
P.
B ut we used a w en-ordered set W
enlarged the class of Euclidean rings [51.
fonowing 閻 ;
as the range of a E uclidean
Samuel asked whether the passage from N to a we11-ordered set W
F or the qustion, M .
N agata proved
the
こ
Theye is a EMdidea11 r緬g R such thd (i) R is aM 紬 t昭 rd doma泌 alld (ii) か r 皿 y
m呻 で : 7? →
N ca11 Mot bc a EMdideaM r泌g.
A N OT E ON E U CLI DE A N RI N G
15
T herefore, our theorem shows that w e got an eχample such that it is not an integral domain.
(2)
A characteristic of a Euclidean ring isthat it isprincipal ideal ring. A ring
in which every finitely generated ideal is principal is called a B6zout dom ain.
N .I .
Dubrovin showed that, under a condition, a projed ive limit of B6zout domain is a
B6zout domain [11。
R eferences
田
N,I. Dubrovin, The projedive limit of rings with elementary divisors, Math. USSR Sbornik 47
(1984), 85- り0.
1 1 1
3 E
4 E
r
a
E
[2] C. R . Fletcher, E uclidean Rings, J. L ondon M ath. Soc. , 4 ( 1971),
79- 82.
K . M ahler, lntroduction to p-adic numbers and their functions, Cambridge U niversity Press, 1973.
M . N agata, 0 n E uclid A lgorithm, T ata lnst. Fund. Res. Studies in M ath. , 8 ( 1978),
P. Sam ue1, A bout E uclidean R ings, J. of A lgebra, 19 ( 1971) ,
85 - 90.
175- 186.

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