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Title
On the Euclidean Modules
Author(s)
AMANO, Kazuo
Citation
[岐阜大学教養部研究報告] vol.[27] p.[63]-[66]
Issue Date
1991
Rights
Version
岐阜大学教養部 (Department of Mathematics, Faculty of
General Education, Gifu University)
URL
http://repository.lib.gifu-u.ac.jp/handle/123456789/47688
※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。
Bulletin of the F aculty of General E ducation, Gifu U niv. vol. 27 ( 1991)
63
On the E uclidean M odules
K azuo A M A N O
D申出 ma t of Matkemd cs
( Received October 16,1991)
1. 1ntroduction.
L et R be a ring, M a left R, module and W a wen ordered set. ln
[21, H.W .Lenstra,Jr constructedthefollowingideaofeuclideanmodules. Letf : M- { O}
→IV be a map. lf for all a, bE M , b≠0, there exist qE R and c E M such that a =
qb十
c, c= Oor f( c) < f( b) , M iscalled left euclideanmodule. Thisisageneralizationof usual
euclidean ring, because a ring R is left euclidean if it is euclidean as a left module over
itself. H e proved the followings;
1. L et M be an euclidean left R-m odule and N ⊂ M be a subm odule. T hen N =
Rχ
for some x E N . [ 2. T h.1.61
2. L et R be a commutative ring and M a left R -M odule. lf χE M satisfies M =
Rχ,
then M is oud dean if and only if the ring R /A nn( x) is euclidean. [ 2. Prop.3.1]
On the other hand, G.E .CQoke introduced the concept of ω-stage euclidean domain
in [11. 1n [11, heprovedthefollowings;
3. L et R be a ring of algebraic integers. lf R is ω-stage euclidean, then it is k-stage
euclidean for some k. [ 1. Prop. 131
4. L et R be an integral dom ain. T hen R is ω-stage euclidean if and only if R is GE 2
and Bezout. [ 1. Prop.141
5. A number ring which is GE2 and has class number l is k-stage euclidean for some
k. [1. Th.1]
ln the present paper, we shall constract the concept of ω, stage euclidean module and
we shall obtain some analogous results to the above properties.
2.
ω-stage euclidean.
L et R be a ring.
M a left R -module and W a w ell ordered
set.
Definition 1. L et a,b be elements of M . A k-stage division chain starting from the
pair ( a,b) is a sequence of equations
64
K azuo A M A N O
a
(D)
b
= q2r1 + r2
r1
二 q3r2 + r3
●
●
●
●
●
●
●
rk- 2= qkrk- 1 + rk
where qlE R and rlE M . Such a division chain is called terminating if the last rem ainder
rh is 0 .
D efinition 2. L et f : M - { O} → W bea map. M is an ω-stageeuclidean modulewith
resped to the m ap f if for every pair ( a,b) w ith b ≠ 0, there eχists a k-stage division
chain ( D) for somek such that thelast remainder rhsatisfies八= Oor f( rj ) く f( b) .
Proposition1. 訂 is a
ω-s収 e a d泌皿 mo加 le if 皿 d o械y が evey:y j)峨
( a, b)
犯池 b≠ O k s αteym油d 昭恥st昭e d屈si回 ck 油加y somek.
PyOOf . lt issimilar tothg proof of Prop.1 1n [21.
D efinition 3. lf for a,bE M there exists qE R such that a= qb, b is caned a divisor
of a, F or thepair ( a,b) , if there eχist c E M and q, rE such that a = qc, b= rc, c iscalled
a common divisor of a and b. lf cis a common divisor and any common divisor of a and
b divides c, c is called a g.c.d. for the pair ( a,b) .
P ropositioU 2 .
jlf tk 卸 iy ( a, b) k s α teym緬 αt泌 g d加磁 ou c辿泌 , 仇四 tk
m xt-to・
㎞ t 粍琲α
泌dey 几_ 11s a g.c、d. 知y ( α
湊) .
PyOOjl Seetheproof of Prop. 3 1n [21 .
N ow , we consider a N eatherian R-module M ( i.e. every ascending chain of its
submodules has only a finite number of distinct terms) . T hen, we have
Proposition 3. L d M be a11 (a-stαge eMdide皿 , χeαlheyia?1
, R一
mod㎡e α11d let N ⊂ が
be α sMbmod㎡e、T k ll N こ Rχ加 y some χモ N 。
PyOOyl SinceM isNeaterian R-module, a submoduleN isfinitelygenerated. Let al,
a 2,
‥ ゛, a n b e a set o f g en er a t o r s o f N . A cco r d in g t o P r o p . 1 a n d P r o p . 2 , P a ir
( a 1, a 2)
h as
a terminating division chain. H encethereexists a g.c.d. cl for ( al,a2) . N ow work cl and
a3, and so on. T hen we have a g.c.d。χ for a1,a2, ‥・,an in N . Clearly we have N = Rχ。
Corollary. Le日 E N sud tk け (y) = g 加 { / ( z) ; Zモ N べ O} } . 毀 a , 五y⊂ N 、
CoroUary. 仔 M・iS l -s恒gea4dideau, Ry= N.
PyOOy
l SinceM is 1-stageeuclidean, thereexist qE R and rE M suchthat x = qy十
On the euclidean modules
65
r, r= Oor f( r) < f( y) . Thenr= x- qyE N sof( r) < f( y) isexcludedbyminimality of f(y) .
H ence r = O and χ= qyE R y.
3. l nequality condition. A ssume that for any a E M - { O} the m ap f has inequality
f( a) ≦f ( qa) for all qe Rづ O} .
Proposition4. が qモ R iS観 itご ( ,2) = / ( 卯) .
PyOO
jl Put b= qa. Sincea = q-1b, f(b) ≦f(q-1b) ≦f( aj
D efinition4.
yE N su(ホ that f ( y) = min { f( z) ; zE N 一 { 0} } is called a minim al
element of N w ith respect to the map f.
Proposition 5.
£ d 訂加加 ω一
s旨ge 四 d i& α肴, N eαtk γ紬R R -mod㎡e αRd ld N ⊂ μ
be α s油 mod㎡e, N is ga em td
by α m細加沢山 me戒 y, if 皿 d ol雨が y is αg, cj 加 y
α set of gem m toys of N .
PyOOjl Let x ag.c.d. for aset of generatorsof N. Since N = Rχ= Ry, ydividesχand
x divides y. H ence y is a g.c.d. for a set of generators of N . Converse is trivia1.
ln the rest of this section we assunie that M is ω-stage euclidean, N eatherian
R-module with respect to a map f and a minim al element of a submodule N is a g.c.d. for
a set of generators of N .
Proposition 6. F Oy χE M 一 { θ} バ d g ( x) = 必 7z { / ( 2) 7 zE 拓 ; 一 { O} } .
Thm M is a筒。一
s短ge 四 d de四面伍 yesj)ed to tk 琲司) g。
PyOOyl For all a, bE M , b≠ 0 , wechooseq E R such that g(d) = f( qb) . SinceM is
ω-stage euclidean. for thd pair ( a, qb)
there exist k -stage division chain ( D )
for som e
k such that the last remainder rl satisfies r1, = O or f( rj ) < f( qb) T hen g( rh) ≦ f( rj )
< f( qb) = g( b) as required.
Proposition 7.
The m的 g sd sjies 緬 du l雨 condiH回 .
PyOO
jl Obvious.
Proposition 8. Ld bE M be a 面戒 md 山 ma t of M 屈 th y叫 ecH o the m呻 g n d
ld A 11筧( ろ) bd hd φ ided
リ E R :ybこ叫 of R . T ha R ZA u11( ろ) is all (・) -stage eMdideα11
j?一
mod14鼠
PyOO
jl For all qE R- { O} wehaveg( qb) ≧g(b) , sowecanwriteg( qb) = g(b) 十t( q)
for some map t : R 一 { O} → W . lf q- q E A nn( b) , t( q) = t( q ) . So t induces a map u : R /
Ann(b) →W . For thepair ( pb, qb) withqb≠O thereexistsak-stagedivisionchain ( D)
for some k such that the last rem ainder rl satisifies r1, = O or g凪 ) < g( qb) . 0 n the oher
66
K azuo A M A N O
hand, ΓμΞRb for all i, so w e can write rl= sib for some slE R . H ence we have a following
k- stage division chain as required in R -module R /A nn( b) ;
-
-
-
p
-
qlq +
-
-
S1
-
q = q2Sl + S2
●
●
●
● ● ●
sk- 2 =
qksk- 1 +
sk,
乱= 百〇
ru(λ) < u(亙), because, u(瓦) = t(sj) = g(sふ) - g(b) < g(qb) - g(b) = t(q) = u滝)
CoroUary. L d R be α commM短t加e yillg. Tk 11 M is 。 -stage eMd ideα箆汀 α肴d o戒 y
汀版画 g RyAm1( ろ) 台 ω-s恒涙 eMdideα歓
references
[1] G.E.Cooke, A weakening of thetheeuclideanproperty for integral domainsandapplicaiionsto
algebraic number theory.I , J. reine angew . M ath 282 ( 1976) , 133-156
[2] L.K.Hua, lntroduction toNumber Theory, Springer-Verlag, 1982
[3] H.W .Lenstra,Jr, LeduresonEuclideanRings, Bielefeld, 1974
[4] W.Narkiewicz, Elementary andAnalyticTheory of AlgebraicNumbers, Springer-Verlag. 1989
[5] P.Samue1, About euclideanrings, J. Algebra 19( 1971) , 282・301

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