2-Simple ore domains of stable rank 1
It is known that a simple Bézout domain is a domain of elementary divisors if and only if it is 2-simple. We prove that, over a 2-simple Ore domain of stable rank 1, an arbitrary matrix that is not a divisor of zero is equivalent to a canonical diagonal matrix.
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| Дата: | 2010 |
|---|---|
| Автори: | , , , |
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| Мова: | Українська Англійська |
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Institute of Mathematics, NAS of Ukraine
2010
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508974936227840 |
|---|---|
| author | Domsha, O.V. Zabavskii, B. V. Домша, О. В. Забавський, Б. В. |
| author_facet | Domsha, O.V. Zabavskii, B. V. Домша, О. В. Забавський, Б. В. |
| author_sort | Domsha, O.V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:41:19Z |
| description | It is known that a simple Bézout domain is a domain of elementary divisors if and only if it is 2-simple. We prove that, over a 2-simple Ore domain of stable rank 1, an arbitrary matrix that is not a divisor of zero is equivalent to a canonical diagonal matrix. |
| first_indexed | 2026-03-24T02:33:44Z |
| format | Article |
| fulltext |
UDK 512.552.12
O. V. Domßa, B. V. Zabavs\kyj (L\viv. nac. un-t im. I. Franka)
2-PROSTI OBLASTI ORE STABIL|NOHO RANHU 1
It is known that the simple Bezout domain is a domain of elementary divisors if and only if it is 2-simple
domain. We prove that, over the 2-simple Ore domain of stable rank 1, an arbitrary matrix that is not a
divizor of zero is equivalent to a canonical diagonal matrix.
Yzvestno, çto prostaq oblast\ Bezu qvlqetsq oblast\g πlementarn¥x delytelej tohda y tol\-
ko tohda, kohda ona 2-prostaq. V rabote dokazano, çto nad 2-prostoj oblast\g Orπ stabyl\noho
ranha 1 proyzvol\naq matryca, ne qvlqgwaqsq delytelem nulq, πkvyvalentna kanonyçeskoj
dyahonal\noj matryce.
Zadaça pro diahonalizacig matryc\ nad kil\cqmy [ klasyçnog. }] prototypom [
teorema Haussa pro ekvivalentnist\ matryci nad polem diahonal\nij matryci z
odynycqmy ta nulqmy na holovnij diahonali. Perßi rezul\taty takoho typu
wodo cilyx çysel buly otrymani v 1861 r. H.:Smitom [1]. Vin doviv, wo koΩna
matrycq z ciloçyslovymy elementamy ßlqxom elementarnyx peretvoren\ rqd-
kiv i stovpciv zvodyt\sq do diahonal\noho vyhlqdu, do toho Ω koΩen diahonal\-
nyj element [ dil\nykom nastupnoho. Pizniße teoremu Smita bulo poßyreno na
rizni klasy kilec\. Tak, Dikson [2], Vedderbarn [3], van der Varden [4] i DΩekob-
son [5] poßyryly danu teoremu na rizni klasy komutatyvnyx i nekomutatyvnyx
kilec\, a Tejxmgller [6] oderΩav povnyj rozv’qzok dlq nekomutatyvnyx ob-
lastej holovnyx idealiv (a v inßomu formulgvanni — Asano [7]).
Vsi ci rezul\taty spryqly vvedenng Kaplans\kym ponqttq kil\cq elemen-
tarnyx dil\nykiv. Nahada[mo, wo matrycq nad asociatyvnym kil\cem z odynyceg
ma[ kanoniçnu diahonal\nu redukcig, qkwo ]] moΩna zvesty do diahonal\noho
vyhlqdu ßlqxom domnoΩennq zliva i sprava na deqki oberneni matryci vidpovid-
nyx rozmiriv, i pry c\omu koΩen diahonal\nyj element [ povnym dil\nykom na-
stupnoho. Qkwo koΩna matrycq nad kil\cem ma[ kanoniçnu diahonal\nu reduk-
cig, to take kil\ce nazyva[t\sq kil\cem elementarnyx dil\nykiv [8].
Qkwo doslidΩennq komutatyvnyx kilec\ elementarnyx dil\nykiv velysq
dosyt\ systematyçno [9 – 12], to nekomutatyvni kil\cq elementarnyx dil\nykiv
doslidΩuvalysq frahmentarno [13, 14]. Krim navedenyx rezul\tativ vartyj uva-
hy rezul\tat Kona [15], qkyj doviv, wo prava holovna oblast\ Bezu [ kil\cem
elementarnyx dil\nykiv. U roboti [16] pobudovano pryklad tako] oblasti Bezu,
pryçomu zauvaΩymo, wo ce pryklad prosto] oblasti Bezu. Sered najnovißyx re-
zul\tativ varto vidmityty [17], de pokazano, wo prosta oblast\ Bezu [ oblastg
elementarnyx dil\nykiv todi i til\ky todi, koly vona [ 2-prostog.
Odnym iz novyx ponqt\, qke vvijßlo v teorig kilec\ z K-teori] i vyqvylos\
korysnym pry rozv’qzanni nyzky vidkrytyx zadaç teori] kilec\, [ ponqttq sta-
bil\noho ranhu kil\cq. Zokrema, dovedeno, wo stabil\nyj ranh kil\cq elemen-
tarnyx dil\nykiv ne perevywu[ 2 [18].
OtΩe, nexaj R — prosta oblast\. Todi dlq dovil\noho nenul\ovoho elemen-
ta a R∈ otryma[mo R a R = R, tobto isnugt\ elementy u1 , u un2, ,… ; v1 ,
v v2, ,… n ∈ R taki, wo
u a1 1v + u a2 2v + … + u an nv = 1.
Qkwo dlq koΩnoho nenul\ovoho elementa a R∈ isnu[ natural\ne çyslo n ta-
ke, wo u a1 1v + … + u an nv = 1, do toho Ω çyslo n [ najmenßym z usix moΩly-
vyx, to oblast\ R nazyva[t\sq n-prostog. Zokrema, oblast\ R [ 2-prostog
todi i til\ky todi, koly dlq dovil\noho nenul\ovoho elementa a R∈ isnugt\
© O. V. DOMÍA, B. V. ZABAVS|KYJ, 2010
1436 ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 10
2-PROSTI OBLASTI ORE STABIL|NOHO RANHU 1 1437
elementy u1 , u2 ; v1 , v2 ∈ R taki, wo u a1 1v + u a2 2v = 1. Prykladom 2-prosto]
oblasti Ore [ kil\ce vid n-dyferencigvan\ [19].
TverdΩennq 1. Nexaj R — 2-prosta oblast\. Todi dlq dovil\nyx nenul\o-
vyx elementiv a, b R∈ isnugt\ elementy u1 , u2 ; v1 , v2 ∈ R taki, wo
u a1 1v + u b2 2v = 1.
Dovedennq. Za umovog teoremy elementy a, b [ nenul\ovymy i R — ob-
last\, todi ab ≠ 0 i
RabR = R .
Oskil\ky R [ 2-prostog oblastg, to isnugt\ elementy x1, x2 ; y1 , y R2 ∈
taki, wo
x aby1 1 + x aby2 2 = 1.
Poklademo x u1 1= , by1 1= v , x a u2 2= , y u2 2= i otryma[mo
u a1 1v + u b2 2v = 1,
wo j potribno bulo dovesty.
Poznaçymo çerez U R( ) hrupu obernenyx elementiv oblasti R .
Nahada[mo, wo kil\ce R [ kil\cem stabil\noho ranhu 1, qkwo z toho, wo
aR + bR = R dlq dovil\nyx elementiv a, b R∈ , vyplyva[ isnuvannq elementiv
t R∈ i u U R∈ ( ) takyx, wo a + bt = u [20].
TverdΩennq 2. Nexaj R — 2-prosta oblast\ stabil\noho ranhu 1. Todi
dlq dovil\nyx nenul\ovyx elementiv a , b R∈ isnugt\ elementy α , β ∈ R i
obernenyj element w U R∈ ( ) taki, wo aα + w bβ = 1.
Dovedennq. Oskil\ky R — 2-prosta oblast\, to zhidno z tverdΩennqm:1
dlq dovil\nyx nenul\ovyx elementiv a, b R∈ isnugt\ taki elementy u1 , u2 ;
v1 , v2 ∈ R , wo
u a1 1v + u a2 2v = 1,
zvidky
u aR1 + u aR2 = R .
Oskil\ky R — oblast\ stabil\noho ranhu 1, to isnu[ takyj element x R∈ ,
dlq qkoho
u a1 + u bx2 ∈ U R( ).
Zvidsy Ra + R bx = R .
Z toho, wo R — oblast\ stabil\noho ranhu 1, otrymu[mo
a + y bx = u ∈ U R( )
dlq deqkoho elementa y R∈ . OtΩe, a R + bR = R .
Znovu Ω z toho, wo R — oblast\ stabil\noho ranhu 1, ma[mo
as + y = w ∈ U R( )
dlq deqkoho elementa s R∈ . Zvidsy
y = w – a s.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 10
1438 O. V. DOMÍA, B. V. ZABAVS|KYJ
Todi rivnist\ a + y bx = u pry otrymanomu y nabere vyhlqdu
a + w bx – a sbx = a (1 – sbx) + w bx = u.
Zvidsy oderΩymo
a sbx u( )1 1− − + wbxu−1 = 1,
tobto
aα + w bβ = 1
dlq deqkyx elementiv α, β ∈ R i obernenoho elementa w U R∈ ( ).
Oznaçennq 1 [21]. Oblast\ R nazyva[t\sq pravog (livog) oblastg Ore,
qkwo dlq dovil\nyx nenul\ovyx elementiv a , b R∈ aR bR∩ ≠ 0 ( Ra Rb∩ ≠
≠ 0{ } ). Oblast\ Ore — ce oblast\, qka [ pravog i livog oblastg Ore odno-
çasno.
Oznaçennq 2 [21]. Matryci A i B nazvemo ekvivalentnymy nad oblastg
R, qkwo isnugt\ oberneni matryci P i Q nad R vidpovidnyx rozmiriv taki,
wo B = PA Q.
Dovedemo nastupne tverdΩennq.
TverdΩennq 3. Nexaj R — 2-prosta oblast\ Ore stabil\noho ranhu 1.
Todi dlq koΩno] matryci A aij= ( ) druhoho porqdku, qka ne [ dil\nykom nulq,
isnugt\ rqdok ( , )1 u i stovpçyk
e
f
taki, wo
( , )1 u A
e
f
= 1,
de e, f R∈ , u U R∈ ( ).
Dovedennq. Oskil\ky R — oblast\ Ore i matrycq A ne [ dil\nykom nulq,
to lehko baçyty, wo isnu[ matrycq D nad R porqdku 2, qka takoΩ ne [
dil\nykom nulq, taka, wo vykonu[t\sq rivnist\
A D =
d
d
1
2
0
0
,
de d1 0≠ , d2 0≠ .
Zhidno z tverdΩennqm:2 dlq elementiv d1 , d2 isnugt\ elementy u, c, d, do
toho Ω u — obernenyj element, taki, wo vykonu[t\sq rivnist\
d c d d1 2 1+ = .
Todi
( )1 a AD
c
d
= ( )1
0
0
1
2
u
d
d
c
d
= 1.
Poklademo
D
c
d
=
e
f
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 10
2-PROSTI OBLASTI ORE STABIL|NOHO RANHU 1 1439
wo dovodyt\ dane tverdΩennq.
Teorema 1. Nexaj R — 2-prosta oblast\ Ore stabil\noho ranhu 1. Todi
dlq dovil\no] matryci A porqdku n, qka ne [ dil\nykom nulq, isnugt\ taki
oberneni matryci P, Q vidpovidnyx rozmiriv, wo
PAQ =
1 0 0 0
0 1 0 0
0 0 0 0
0 0 1 0
0 0 0
…
…
�
…
… ∆
.
Dovedennq provodymo metodom matematyçno] indukci]. Rozhlqnemo vypadok
n = 2. Zhidno z tverdΩennqm:3 dlq matryci A isnugt\ elementy u, e, f R∈ , do
toho Ω u — obernenyj element, taki, wo
( )1 u A
e
f
= 1.
Oçevydno, wo R e + R f = R . Oskil\ky R — oblast\ stabil\noho ranhu 1, to, vra-
xovugçy [5], rqdok ( )1 u i stovpçyk
e
f
moΩna dopovnyty do obernenyx
matryc\ P i Q vidpovidno. Zvidsy
PAQ =
1 ∗
∗ ∗
.
Oçevydno, wo elementarnymy peretvorennqmy rqdkiv i stovpçykiv matrycq
PAQ zvodyt\sq do vyhlqdu
1 0
0 ∆
.
Ce oznaça[, wo dlq matryci A isnugt\ taki matryci S i T, wo
SA T =
1 0
0 δ
.
Nexaj n = 3, tobto matrycq A = ( )aij [ matryceg tret\oho porqdku. Bez obme-
Ωennq zahal\nosti moΩna vvaΩaty, wo pidmatrycq
a a
a a
11 12
21 22
matryci A ne [ dil\nykom nulq. Todi zhidno z tverdΩennqm:3 vykonu[t\sq
rivnist\
1 0
1
13 23
13 23u A
e e a ua
f f a ua( )
− +
− +
( )
( ) = 1.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 10
1440 O. V. DOMÍA, B. V. ZABAVS|KYJ
Oçevydno, wo rqdok 1 0u( ) moΩna dopovnyty do oberneno] matryci P.
ZauvaΩymo, wo R e e a ua− +( )( )13 23 + R f f a ua− +( )( )13 23 = R . Oskil\ky R —
oblast\ stabil\noho ranhu 1, to zhidno z [18] stovpçyk
e e a ua
f f a ua
− +
− +
( )
( )
13 23
13 23
1
moΩna dopovnyty do oberneno] matryci Q. Zvidsy
PA Q =
1 12 13
21 22 23
31 32 33
′ ′
′ ′ ′
′ ′ ′
a a
a a a
a a a
..
Oçevydno, wo matrycq PA Q elementarnymy peretvorennqmy rqdkiv i stovpçy-
kiv zvodyt\sq do vyhlqdu
1 0 0
0
0
a b
c d
.
Oskil\ky A ne [ dil\nykom nulq, to matrycq
a b
c d
teΩ ne [ dil\nykom nulq, a otΩe, za dovedenym vywe, zvodyt\sq do vyhlqdu
1 0
0 ∆
.
Todi matrycq A zvodyt\sq do vyhlqdu
1 0 0
0 1 0
0 0 ∆
,
wo j potribno bulo dovesty.
Indukciq za rozmiramy matryci zaverßu[ dovedennq.
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OderΩano 22.06.09,
pislq doopracgvannq — 23.07.10
|
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| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| last_indexed | 2026-03-24T02:33:44Z |
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| spelling | umjimathkievua-article-29682020-03-18T19:41:19Z 2-Simple ore domains of stable rank 1 2-Прості області Оре стабільного рангу 1 Domsha, O.V. Zabavskii, B. V. Домша, О. В. Забавський, Б. В. It is known that a simple Bézout domain is a domain of elementary divisors if and only if it is 2-simple. We prove that, over a 2-simple Ore domain of stable rank 1, an arbitrary matrix that is not a divisor of zero is equivalent to a canonical diagonal matrix. Известно, ч то простая область Безу является областью элементарных делителей тогда и только тогда, когда она 2-простая. В работе доказано, что над 2-просгой областью Орэ стабильного ранга 1 произвольная матрица, не являющаяся делителем нуля, эквивалентна канонической диагональной матрице. Institute of Mathematics, NAS of Ukraine 2010-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2968 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 10 (2010); 1436–1440 Український математичний журнал; Том 62 № 10 (2010); 1436–1440 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2968/2686 https://umj.imath.kiev.ua/index.php/umj/article/view/2968/2687 Copyright (c) 2010 Domsha O.V.; Zabavskii B. V. |
| spellingShingle | Domsha, O.V. Zabavskii, B. V. Домша, О. В. Забавський, Б. В. 2-Simple ore domains of stable rank 1 |
| title | 2-Simple ore domains of stable rank 1 |
| title_alt | 2-Прості області Оре стабільного рангу 1 |
| title_full | 2-Simple ore domains of stable rank 1 |
| title_fullStr | 2-Simple ore domains of stable rank 1 |
| title_full_unstemmed | 2-Simple ore domains of stable rank 1 |
| title_short | 2-Simple ore domains of stable rank 1 |
| title_sort | 2-simple ore domains of stable rank 1 |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2968 |
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