Singularities of the structure of two-sided ideals of a domain of elementary divisors
We prove that, in a domain of elementary divisors, the intersection of all nontrivial two-sided ideals is equal to zero. We also show that a Bézout domain with finitely many two-sided ideals is a domain of elementary divisors if and only if it is a 2-simple Bézout domain.
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| Date: | 2010 |
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| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2010
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2916 |
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| author | Bilyavs’ka, S. I. Zabavskii, B. V. Білявська, С. І. Забавський, Б. В. |
| author_facet | Bilyavs’ka, S. I. Zabavskii, B. V. Білявська, С. І. Забавський, Б. В. |
| author_sort | Bilyavs’ka, S. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:40:12Z |
| description | We prove that, in a domain of elementary divisors, the intersection of all nontrivial two-sided ideals is equal to zero. We also show that a Bézout domain with finitely many two-sided ideals is a domain of elementary divisors if and only if it is a 2-simple Bézout domain. |
| first_indexed | 2026-03-24T02:32:46Z |
| format | Article |
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K O R O T K I P O V I D O M L E N N Q
UDK 512.552.12
S. I. Bilqvs\ka, B. V. Zabavs\kyj (L\viv. nac. un-t)
OSOBLYVOSTI STRUKTURY DVOBIÇNYX IDEALIV
OBLASTI ELEMENTARNYX DIL|NYKIV
We prove that, in a domain of elementary divisors, the intersection of all nontrivial two-sided ideals is
equal to zero. We also show that the Bezout domain with a finite number of two-sided ideals is a
domain of elementary divisors if and only if it is the 2-simple Bezout domain.
Dokazano, çto v oblasty πlementarn¥x delytelej pereseçenye vsex netryvyal\n¥x dvustoron-
nyx ydealov ravno nulg. TakΩe pokazano, çto oblast\ Bezu s koneçn¥m çyslom dvustoronnyx
ydealov qvlqetsq oblast\g πlementarn¥x delytelej tohda y tol\ko tohda, kohda ona est\ 2-
prostaq oblast\ Bezu.
U roboti [1] opysano prosti oblasti elementarnyx dil\nykiv qk 2-prosti oblasti
Bezu. Krim toho, v [2] otrymano analohiçnyj rezul\tat pro majΩe prosti oblas-
ti elementarnyx dil\nykiv. U danij roboti ci rezul\taty poßyreno na vypadok
oblastej Bezu zi skinçennym çyslom dvobiçnyx idealiv, a takoΩ pokazano, wo
peretyn netryvial\nyx dvobiçnyx idealiv oblasti elementarnyx dil\nykiv doriv-
ng[ nulg.
Pid kil\cem R rozumitymemo asociatyvne kil\ce z 1 ≠ 0 bez dil\nykiv nulq,
a pid prostym kil\cem — kil\ce, v qkomu isnugt\ lyße tryvial\ni dvobiçni idea-
ly { }0 i R .
Nexaj R — proste kil\ce. Todi dlq dovil\noho nenul\ovoho elementa a ma-
[mo RaR = R . Zvidsy vyplyva[, wo isnugt\ elementy u u uk1 2, , , ,… v v1 2, , …
… ∈, vk R taki, wo
u a u a u ak k1 1 2 2v v v+ + … + = 1.
Qkwo dlq vsix nenul\ovyx elementiv a R∈ isnu[ natural\ne çyslo n take, wo
u a u a u an n1 1 2 2v v v+ + … + = 1 dlq deqkyx elementiv u u un1 2 1 2, , , , , ,… …v v
… ∈, vn R , do toho Ω çyslo n [ najmenßym z usix moΩlyvyx, to kil\ce R na-
zyva[t\sq n-prostym. U roboti [3] pokazano, wo kil\ce matryc\ M Pn ( ) , de P
— pole, [ n-prostym, ale ne [ ( )n − 1 -prostym. Prykladom 2-prosto] oblasti [
dyferencial\ne kil\ce vid n dyferencigvan\ [1]. ZauvaΩymo, wo pry n = 1
kil\ce vid odnoho dyferencigvannq [ prykladom 2-prosto] oblasti holovnyx
idealiv. Prykladom 1-prosto] oblasti moΩe buty neskinçenne proste kil\ce [4].
Nahada[mo, wo majΩe prosta oblast\ — ce oblast\ z wonajbil\ße odnym ne-
tryvial\nym dvobiçnym idealom, qkyj zbiha[t\sq z radykalom DΩekobsona [2].
Pid oblastg elementarnyx dil\nykiv rozumitymemo oblast\ R , v qkij dovil\na
( )n m× -matrycq A ma[ kanoniçnu diahonal\nu redukcig, tobto dlq qko] isnu-
gt\ oborotni matryci P GL Rn∈ ( ) , Q GL Rm∈ ( ) taki, wo
© S. I. BILQVS|KA, B. V. ZABAVS|KYJ, 2010
854 ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
OSOBLYVOSTI STRUKTURY DVOBIÇNYX IDEALIV OBLASTI … 855
P A Q =
ε
ε
1 0 0
0 0
0
0 0
0 0
0 0 0
… … …
…
… …
… … …
� � �
� � � � �
� �
� � � �
� � � �
n
,
de R R R Ri i iε ε ε+ ⊆1 ∩ [5].
ZauvaΩymo, wo çerez GL Rn ( ) poznaçeno povnu linijnu hrupu porqdku n nad
kil\cem R .
Teorema)1. V oblasti elementarnyx dil\nykiv peretyn usix netryvial\nyx
dvobiçnyx idealiv dorivng[ nulg.
Dovedennq. Nexaj N = Iα∩ , de { }Iα — mnoΩyna vsix netryvial\nyx
dvobiçnyx idealiv R . Nexaj N ≠ ( )0 i element a N∈ \{ }0 . Oskil\ky R — ob-
last\ elementarnyx dil\nykiv, to dlq matryci
a
a
0
0
isnugt\ oborotni
matryci P p GL Rij= ∈( ) ( )2 ta Q q GL Rij= ∈( ) ( )2 taki, wo
a
a
P
0
0
= Q
z
b
0
0
, (1)
de RbR zR Rz⊆ ∩ . Todi RzR N⊆ , tomu wo RaR = RzR RbR+ = RzR .
ZauvaΩymo, wo z ≠ 0, bo v protyleΩnomu vypadku z (1) vyplyva[ RbR =
= ( )0 , a otΩe b = 0, wo nemoΩlyvo. Vnaslidok toho, wo RbR RzR⊆ i
RzR N⊆ , za oznaçennqm dvobiçnoho idealu N moΩlyvi lyße nastupni vypad-
ky:
1) RbR = { }0 ;
2) RbR RzR N= = .
ZauvaΩymo, wo qkwo RbR = ( )0 , to b = 0. Iz (1) vyplyva[ ap12 = 0 i
ap22 = 0. Za prypuwennqm a ≠ 0, i oskil\ky R — oblast\, to p12 = p22 = 0,
wo nemoΩlyvo, oskil\ky ci elementy [ elementamy druhoho stovpçyka oborot-
no] matryci.
Rozhlqnemo druhyj vypadok. Nexaj RbR = RzR . Todi ma[ misce vklgçen-
nq
RzR = RbR = zR Rz∩ ⊆ RzR .
Takym çynom, RzR = zR i RzR = Rz . Zvidsy z R = R z .
Rozhlqnemo element z2
, todi z R2 = Rz2 ⊂ RzR = Rz = zR . OtΩe,
z R2 ⊂ N. Vraxovugçy vyraz dlq N, ma[mo z R2 = R = zR . Oskil\ky R —
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
856 S. I. BILQVS|KA, B. V. ZABAVS|KYJ
oblast\, to ce moΩlyvo lyße u vypadku, koly z — zvorotnyj element abo z =
= 0, wo nemoΩlyvo vnaslidok vyboru elementa a .
Teorema)2. Oblast\ Bezu zi skinçennym çyslom dvobiçnyx idealiv [ oblastg
elementarnyx dil\nykiv todi i til\ky todi, koly vona [ 2-prostog oblastg
Bezu.
Dovedennq. Nexaj R — oblast\ elementarnyx dil\nykiv zi skinçennym çys-
lom dvobiçnyx idealiv. MoΩlyvi dva vypadky:
1) R — neprosta oblast\ Bezu;
2) R — prosta oblast\ Bezu.
Rozhlqnemo perßyj vypadok, todi v R isnu[ skinçenne çyslo dvobiçnyx idea-
liv N N Nn1 2, , ,… . Nexaj N = Nii
n
=1∩ . ZauvaΩymo, wo oskil\ky R — ob-
last\, to N ≠ ( )0 , a ce za teoremogG1 nemoΩlyvo. OtΩe, perßyj vypadok
nemoΩlyvyj.
Nexaj R [ prostog oblastg elementarnyx dil\nykiv i a — dovil\nyj nenu-
l\ovyj element oblasti R . Oskil\ky R — oblast\ elementarnyx dil\nykiv, to
dlq matryci
a
a
0
0
isnugt\ oborotni matryci P p GL Rij= ∈( ) ( )2 , Q =
= ( ) ( )q GL Rij ∈ 2 ta matrycq
z
b
0
0
taki, wo ma[ misce (1), do toho Ω
RbR zR Rz⊆ ∩ . Oskil\ky R — prosta oblast\, to moΩlyvi dva vypadky:
b = 0 abo z — zvorotnyj element R . Qkwo b = 0, to zhidno z (1) ma[mo
ap12 = 0, ap22 = 0. (2)
Vnaslidok toho, wo R — oblast\ i a ≠ 0, rivnist\ (2) moΩlyva lyße u vypad-
ku, koly p12 = p22 = 0, wo nemoΩlyvo, oskil\ky P GL R∈ 2( ) .
OtΩe, moΩlyvym [ lyße vypadok, koly z — zvorotnyj element R . Z toç-
nistg do ekvivalentnosti matryc\ moΩemo vvaΩaty, wo z = 1. Todi z (1) otry-
mu[mo
ap11 = q11 , ap21 = q21. (3)
Matrycq Q [ oborotnog, todi Rq Rq11 21+ = R , zvidky uq uq11 21+ = 1
dlq deqkyx elementiv u, v ∈ R . Todi z rivnosti (3) otrymu[mo uap ap11 21+ v = 1,
tobto element a [ 2-prostym. Vnaslidok dovil\nosti nenul\ovoho elementa a
neobxidnist\ dovedeno.
Z uraxuvannqm [1] dostatnist\ [ oçevydnog.
1. Zabavskyj B. V. Prost¥e kol\ca normal\n¥x delytelej // Mat. stud. – 2004. – 22, # 2. –
S.G129 – 133.
2. Zabavsky B. V., Kysil T. N. Nearly simple elementary divisor domain // Bull. Acad. Stiinte Rep.
Mold. Mat. – 2006. – # 3. – P. 121 – 123.
3. Olszewski J. On ideals of products of rings // Demonst. math. – 1994. – 27, # 1. – P. 1 – 7.
4. Lam T., Dugas A. Quasi-duo rings and stable range descent // J. Pure and Appl. Algebra. – 2005. –
195. – P. 243 – 259.
5. Kaplansky I. Elementary divisors and modules // Trans. Amer. Math. Soc. – 1949. – 66 . –
P. 464 – 491.
OderΩano 23.09.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
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| id | umjimathkievua-article-2916 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:32:46Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e9/1d8d045667e32349222894bf8de1f3e9.pdf |
| spelling | umjimathkievua-article-29162020-03-18T19:40:12Z Singularities of the structure of two-sided ideals of a domain of elementary divisors Особливості структури двобічних ідеалів області елементарних дільників Bilyavs’ka, S. I. Zabavskii, B. V. Білявська, С. І. Забавський, Б. В. We prove that, in a domain of elementary divisors, the intersection of all nontrivial two-sided ideals is equal to zero. We also show that a Bézout domain with finitely many two-sided ideals is a domain of elementary divisors if and only if it is a 2-simple Bézout domain. Доказано, что в области элементарных делителей пересечение всех нетривиальных двусторонних идеалов равно нулю. Также показано, что область Безу с конечным числом двусторонних идеалов является областью элементарных делителей тогда и только тогда, когда она есть 2-простая область Безу. Institute of Mathematics, NAS of Ukraine 2010-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2916 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 6 (2010); 854 – 856 Український математичний журнал; Том 62 № 6 (2010); 854 – 856 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2916/2583 https://umj.imath.kiev.ua/index.php/umj/article/view/2916/2584 Copyright (c) 2010 Bilyavs’ka S. I.; Zabavskii B. V. |
| spellingShingle | Bilyavs’ka, S. I. Zabavskii, B. V. Білявська, С. І. Забавський, Б. В. Singularities of the structure of two-sided ideals of a domain of elementary divisors |
| title | Singularities of the structure of two-sided ideals of a domain of elementary divisors |
| title_alt | Особливості структури двобічних ідеалів області елементарних дільників |
| title_full | Singularities of the structure of two-sided ideals of a domain of elementary divisors |
| title_fullStr | Singularities of the structure of two-sided ideals of a domain of elementary divisors |
| title_full_unstemmed | Singularities of the structure of two-sided ideals of a domain of elementary divisors |
| title_short | Singularities of the structure of two-sided ideals of a domain of elementary divisors |
| title_sort | singularities of the structure of two-sided ideals of a domain of elementary divisors |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2916 |
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