On the stable range of matrix rings
It is shown that an adequate ring with nonzero Jacobson radical has a stable range 1. A class of matrices over an adequate ring with stable range 1 is indicated.
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2009
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| author | Zabavskii, B. V. Petrichkovich, V. M. Забавський, Б. В. Петричкович, В. М. |
| author_facet | Zabavskii, B. V. Petrichkovich, V. M. Забавський, Б. В. Петричкович, В. М. |
| author_sort | Zabavskii, B. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:45:43Z |
| description | It is shown that an adequate ring with nonzero Jacobson radical has a stable range 1. A class of matrices over an adequate ring with stable range 1 is indicated. |
| first_indexed | 2026-03-24T02:36:40Z |
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| fulltext |
K O R O T K I P O V I D O M L E N N Q
UDK 512.55+512.64
B. V. Zabavs\kyj (L\viv. nac. un-t),
V. M. Petryçkovyç (In-t prykl. prob. mexaniky i matematyky NAN Ukra]ny, L\viv)
PRO STABIL|NYJ RANH KILEC| MATRYC|
We prove that an adequate ring with nonzero Jacobson radical is of stable range one. A class of matrices
over an adequate ring having the stable range one is established.
Dokazano, çto adekvatnoe kol\co s nenulev¥m radykalom DΩekobsona ymeet stabyl\n¥j ranh
odyn. Ukazan klass matryc nad adekvatn¥m kol\com, ymegwyj stabyl\n¥j ranh odyn.
Stabil\nyj ranh [ odnym z osnovnyx invariantiv K-teori]. Ce ponqttq, vvedene
X. Bassom [1], vykorystovu[t\sq v teori] kilec\, zokrema v zadaçax diahonal\no]
redukci] matryc\ [2, 3]. VaΩlyvym pytannqm [ vyvçennq zv’qzkiv stabil\noho
ranhu kil\cq M n R( , ) matryc\ porqdku n nad R i stabil\noho ranhu kil\cq
R . U robotax [4, 5] vstanovleno, wo stabil\nyj ranh r kil\cq matryc\
M n R( , ) dorivng[ 1 + ( )/r n−[ ]1 , de m[ ] oznaça[ cilu çastynu çysla m .
Tomu qkwo stabil\nyj ranh kil\cq R dorivng[ 1 abo 2, to stabil\nyj ranh kil\-
cq matryc\ M n R( , ) dorivng[ vidpovidno 1 abo 2.
Sered kilec\ skinçennoho stabil\noho ranhu slid vydilyty klas kilec\ ele-
mentarnyx dil\nykiv, qkyj bulo vvedeno I. Kaplans\kym [6]. Bil\ßist\ vidomyx
klasiv kilec\ elementarnyx dil\nykiv sutt[vo zaleΩat\ vid umov obryvu zros-
tagçyx lancghiv idealiv. Perßyj pryklad kil\cq elementarnyx dil\nykiv bez
umov obryvu zrostagçyx lancghiv idealiv buv navedenyj Vedderbarnom, a same
takym [ kil\ce analityçnyx funkcij [7]. Cej pryklad dozvolyv O. Xelmeru
vvesty novyj klas kilec\ elementarnyx dil\nykiv, qkyj otrymav nazvu adekvat-
nyx kilec\ [8]. Vidomo [9], wo stabil\nyj ranh adekvatnoho kil\cq ne perevywu[
2. U bahat\ox vypadkax vin dorivng[ 1.
U cij statti vkazano umovy, za qkyx adekvatne kil\ce [ stabil\noho ranhu 1.
Na osnovi standartno] formy pary matryc\ wodo uzahal\neno] ekvivalentnosti u
kil\ci M n R( , ) matryc\ porqdku n nad adekvatnym kil\cem R vydileno klas
matryc\ stabil\noho ranhu 1, koly kil\ce R moΩe buty stabil\noho ranhu
bil\ßoho za 1.
Nexaj R — adekvatne kil\ce, tobto R — oblast\ cilisnosti, v qkij koΩnyj
skinçennoporodΩenyj ideal [ holovnym i dlq koΩnoho nenul\ovoho elementa
a R∈ i koΩnoho elementa b R∈ isnugt\ taki elementy c, d R∈ , wo a = cd,
do toho Ω c [ vza[mno prostym iz b, a koΩnyj neoborotnyj dil\nyk di elemen-
ta d ma[ neoborotnyj spil\nyj dil\nyk iz b [8]. Rqdok a a an1 2 … elemen-
tiv kil\cq R nazyva[t\sq unimodulqrnym (prymityvnym), qkwo a R1 +@ a R2 + …
… + a Rn = R . Stabil\nym ranhom kil\cq R nazyva[t\sq najmenße natural\ne
çyslo m take, wo dlq dovil\noho unimodulqrnoho rqdka a a1 2 … a am m + 1
nad kil\cem R isnugt\ taki elementy b1, b b Rm2, ,… ∈ , wo rqdok a1 +
+ a bm + 1 1 a a bm2 1 2+ + … a a bm m m+ + 1 [ unimodulqrnym. Qkwo takoho natu-
ral\noho çysla ne isnu[, to vvaΩagt\, wo stabil\nyj ranh kil\cq dorivng[ ne-
skinçennosti [3, 4].
© B. V. ZABAVS|KYJ, V. M. PETRYÇKOVYÇ, 2009
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 11 1575
1576 B. V. ZABAVS|KYJ, V. M. PETRYÇKOVYÇ
Teorema 1. Nexaj R — adekvatne kil\ce take, wo joho radykal DΩekobso-
na [ nenul\ovym. Todi stabil\nyj ranh kil\cq R dorivng[ 1.
Dovedennq. Nexaj a, b, c R∈ , do toho Ω c ≠ 0 i a R + bR + cR = R . Todi
c = rs , de rR + aR = R i s R aR1 + ≠ R dlq dovil\noho s1 takoho, wo
sR s R⊂ 1 ≠ R . Zhidno z [9] ( )a br R+ + cR = R . Nexaj J R( ) — nenul\ovyj ra-
dykal DΩekobsona adekvatnoho kil\cq R i a, b R∈ , do toho Ω a R + bR = R .
Todi, vybyragçy dovil\ne c J R∈ ( ) , c ≠ 0, baçymo, wo isnu[ takyj element
r R∈ , wo ( )a br R+ + cR = R, tobto ( )a br u c+ + v = 1. Oskil\ky c J R∈ ( ) ,
to ( )a br u+ = 1 − cv — oborotnyj element kil\cq R, tobto (a + br R) = R .
Teoremu dovedeno.
ZauvaΩymo, wo u c\omu rezul\tati obmeΩennq vidsutnosti dil\nykiv nulq u
kil\ci R [ nesutt[vym.
KoΩna matrycq A M n R∈ ( , ) nad adekvatnym kil\cem R ma[ vlastyvist\
kanoniçno] diahonal\no] redukci], tobto isnugt\ taki oborotni matryci U, V ∈
∈ GL n R( , ) , wo
UAV DA A A
r
A= = … …( )diag µ µ µ1 2 0 0, , , , , , ,
de µr
A ≠ 0 i µ µi
A
i
A
+ 1 , i = 1, 2, … , r – 1. Matrycg DA
nazyvagt\ kanoniçnog
diahonal\nog formog abo normal\nog formog Smita matryci A M n R∈ ( , ) .
Pary matryc\ ( , )A A1 2 i ( , )B B1 2 , Ai , B M n Ri ∈ ( , ) , i = 1, 2, nazyvagt\sq
uzahal\neno ekvivalentnymy, qkwo A UB Vi i i= , i = 1, 2, dlq deqkyx matryc\ U,
V1 , V GL n R2 ∈ ( , ) [10]. Vyvçennq takoho typu ekvivalentnostej par matryc\
potrebugt\ bahato zadaç. Kanoniçni formy wodo tako] ekvivalentnosti pobudo-
vani lyße dlq par matryc\ nad polqmy [11, 12]. U roboti [13] wodo uzahal\ne-
no] ekvivalentnosti vstanovleno standartnu formu ( DA , T TDB B= ) pary mat-
ryc\ ( , )A B , A, B M n R∈ ( , ) , abo ]] vidpovidnu standartnu paru, de T ti j
n
=
1
— nyΩnq unitrykutna matrycq, tobto ti j = 0, qkwo i < j i ti j , i > j, naleΩat\
povnij systemi lyßkiv za modulem δi j =
µ
µ
µ
µ
i
A
j
A
i
B
j
B
,
u vypadku, koly R — adek-
vatne kil\ce.
Nabir matryc\ ( ,A A1 2 , … , Ak ) , A M n Ri ∈ ( , ) , i = 1, 2, … , k, nazyva[mo
prostym (prymityvnym), qkwo A V1 1 + A V2 2 + … + A Vk k = I dlq deqkyx matryc\
V M n Ri ∈ ( , ) , i = 1, 2, … , k, I — odynyçna matrycq.
Çerez ′M R( , )2 poznaçatymemo klas matryc\ ai j 1
2
druhoho porqdku ta-
kyx, wo ( ,a a11 12 , a a21 22, ) = 1.
Lema 1. Nexaj para matryc\ ( , )A B , A , B M R∈ ′( , )2 , [ prostog. Todi
para matryc\ ( , )A B uzahal\neno ekvivalentna do standartno] pary (DA ,
T B ) odnoho iz takyx vyhlqdiv:
1 0
0
1 0
ϕ ψ
,
t
, qkwo rang rangA B= = 2 , (1)
de
t =
=
≠
0 1
1 1
, ( , ) ,
, ( , ) ;
qkwo
qkwo
ϕ ψ
ϕ ψ
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 11
PRO STABIL|NYJ RANH KILEC| MATRYC| 1577
1 0
0 0
1 0
,
t ψ
, qkwo rang A = 1 , rang B = 2 , (2)
de
t
U R
U R
=
∈
∉
0
1
, ( ),
, ( ),
qkwo
qkwo
ψ
ψ
U R( ) — hrupa odynyc\ kil\cq R;
1 0
0 0
0 0
1 0
,
, qkwo rang A = 1 , rang B = 1 . (3)
Dovedennq. Nexaj ( , )A B — prosta para i rang A = rang B = 2. Todi na os-
novi teoremy 1 iz roboty [13] para matryc\ ( , )A B uzahal\neno ekvivalentna do
standartno] pary ( , )D TA B
vyhlqdu
1 0
0
1 0
ϕ ψ
,
t
. Spil\ni livi dil\-
nyky matryc\ DA
i T B
, z toçnistg do pravo] asocijovnosti, magt\ vyhlqd
D di i= diag ( , )1 , de di ϕ i d ti ( , )ψ , tomu wo v c\omu vypadku zhidno iz re-
zul\tatamy roboty [14] dil\nyky iz zadanog kanoniçnog diahonal\nog formog
matryc\ DA
i T B
z toçnistg do pravo] asocijovnosti vyznaçagt\sq odnoznaç-
no. Iz toho, wo para matryc\ ( , )A B [ prostog, vyplyva[, wo i ]] vidpovidna
standartna para ( , )D TA B
[ prostog. Ce oznaça[, wo livi spil\ni dil\nyky
matryc\ DA
i T B
[ tryvial\nymy. Tomu di = 1 i ϕ ψ, ( , )t( ) = 1. Todi na os-
novi teoremy 3 iz roboty [13] standartna forma pary matryc\ ( , )A B ma[ vy-
hlqd (1).
U vypadkax, koly para matryc\ ( , )A B uzahal\neno ekvivalentna do par
matryc\ vyhlqdiv (2) abo (3), pry dovedenni lemy mirku[mo analohiçno.
Lemu dovedeno.
ZauvaΩymo, wo u vypadku, koly rang A = 0, tobto A = 0 — nul\ova matrycq
i para matryc\ ( , )A B [ prostog, oçevydno, wo B — oborotna matrycq i stan-
dartnog parog dlq ( , )A B [ para ( , )0 I .
Teorema 2. Nexaj para matryc\ ( , )A B , A , B M R∈ ′( , )2 , [ prostog,
tobto
AU BV I+ = , U, V M R∈ ( , )2 . (4)
Todi isnu[ matrycq P M R∈ ( , )2 taka, wo
AP B Q+ = , (5)
de Q — oborotna matrycq iz GL R( , )2 .
Dovedennq. Zrozumilo, wo koΩna para matryc\ ( , )A B1 1 , qka uzahal\neno
ekvivalentna do prosto] pary ( , )A B , [ prostog, tobto dlq ne] spravdΩu[t\sq
spivvidnoßennq vyhlqdu (4). Tomu dostatn\o pokazaty, wo iz spivvidnoßennq (4)
vyplyva[ (5) dlq pary matryc\ ( , )D TA B
u standartnij formi.
Nexaj para matryc\ ( , )A B uzahal\neno ekvivalentna do standartno] pary
vyhlqdu (1). Todi iz spivvidnoßennq (5) matymemo D P TA B+ = Q abo
1 0
0
1 01 2
3 4
1 2
3 4ϕ ψ
p p
p p t
q q
q q
+ = , (6)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 11
1578 B. V. ZABAVS|KYJ, V. M. PETRYÇKOVYÇ
de
q q q q1 4 2 3 1– = . (7)
Iz spivvidnoßennq (6) oderΩu[mo rivnosti
q p1 1 1= + , q p2 2= , q p t3 3= +ϕ , q p4 4= +ϕ ψ . (8)
Todi iz rivnostej (7) i (8) otrymu[mo
( ) ( ) ( )p p p p t1 4 2 31 1+ + − + =ϕ ψ ϕ
abo
ϕ ψ( ) ( )p p p p p p t p1 4 4 2 3 1 21 1+ − + + − = . (9)
Nexaj u standartnij pari matryc\ (1) t = 0. Todi ( , )ϕ ψ = 1 i diofantove
rivnqnnq
ϕ ψ( ) ( )p p p p p p1 4 4 2 3 1 1 1+ − + + =
vidnosno pi , i = 1, … , 4, ma[ rozv’qzky nad R .
Nexaj teper t = 1. Todi ( , )ϕ ψ ≠ 1. Poklademo u (9) p2 1= − . Todi iz c\oho
spivvidnoßennq oderΩymo rivnqnnq
ϕ ψ( ) ( )p p p p p1 4 4 3 1 1 0+ + + + = ,
qke ma[ rozv’qzky pi , i = 1, … , 4, nad R .
OtΩe, isnu[ matrycq P taka, wo D P TA B+ = Q — oborotna matrycq.
Dovedennq teoremy u vypadkax, koly para matryc\ ( , )A B uzahal\neno ekvi-
valentna do par matryc\ vyhlqdiv (2) abo (3), provodyt\sq analohiçno.
Teoremu dovedeno.
1. Bass H. K-theory and stable algebra // Publ. Math. – 1964. – 22. – P. 5 – 60.
2. Zabavsky B. V. Diagonalizability theorem for matrices over ring with finite stable range // Algebra
and Discrete Math. – 2005. – # 1. – P. 134 – 148.
3. Zabavsky B. V. Diagonalization of matrices over ring with finite stable range // Visnyk Lviv. Univ.
Ser. Mech.-Math. – 2003. – 61. – P. 206 – 211.
4. Vaserßtejn L. N. Stabyl\n¥j ranh kolec y razmernost\ topolohyçeskyx prostranstv //
Funkcyon. analyz y eho pryl. – 1971. – 5, # 2. – S. 17 – 27.
5. Vaserstein L. N. Bass’s first stable range condition // J. Pure and Appl. Algebra. – 1984. – 34. –
P. 319 – 330.
6. Kaplansky I. Elementary divisor ring and modules // Trans. Amer. Math. Soc. – 1949. – 66. – P. 464
– 491.
7. Wedderburn J. H. M. On matrices whose coefficients are functions of single variable // Ibid. – 1915.
– 169, # 2. – P. 328 – 332.
8. Helmer O. The elementary divisor theorem for certain rings without chain conditions // Bull. Amer.
Math. Soc. – 1943. – 49. – P. 225 – 236.
9. Zabavs\kyj B. V., Komarnyc\kyj M. Q. Teorema koenovoho typu dlq adekvatnosti ta kil\cq
elementarnyx dil\nykiv // Mat. metody i fiz.-mex. polq. – 2008. – 49, # 4. – S. 94 – 98.
10. Petrychkovych V. Generalized equivalence of pairs of matrices // Linear and Multilinear Algebra. –
2000. – 48, # 2. – P. 179 – 188.
11. Dlab V., Ringel C. M. Canonical forms of pairs of complex matrices // Linear Algebra and Appl. –
1991. – 147. – P. 387 – 410.
12. Gaiduk T. N., Sergeichuk V. V. Generic canonical form of pairs of matrices with zeros // Ibid. –
2004. – 380. – P. 241 – 251.
13. Petrychkovych V. Standard form of pairs of matrices with respect to generalized equivalence //
Visnyk Lviv. Univ. Ser. Mech.-Math. – 2003. – 61. – P. 148 – 155.
14. Petryçkovyç V. M. Pro paralel\ni faktoryzaci] matryc\ nad kil\cqmy holovnyx idealiv //
Mat. metody i fiz.-mex. polq. – 1997. – 40, # 4. – S. 96 – 100.
OderΩano 18.03.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 11
|
| id | umjimathkievua-article-3123 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:36:40Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/8a/589afd783d80c1654cbc25aaf3030d8a.pdf |
| spelling | umjimathkievua-article-31232020-03-18T19:45:43Z On the stable range of matrix rings Про стабільний ранг кілець матриць Zabavskii, B. V. Petrichkovich, V. M. Забавський, Б. В. Петричкович, В. М. It is shown that an adequate ring with nonzero Jacobson radical has a stable range 1. A class of matrices over an adequate ring with stable range 1 is indicated. Доказано, что адекватное кольцо с ненулевым радикалом Джекобсона имеет стабильный ранг один. Указан класс матриц над адекватным кольцом, имеющий стабильный ранг один. Institute of Mathematics, NAS of Ukraine 2009-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3123 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 11 (2009); 1575-1578 Український математичний журнал; Том 61 № 11 (2009); 1575-1578 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3123/2994 https://umj.imath.kiev.ua/index.php/umj/article/view/3123/2995 Copyright (c) 2009 Zabavskii B. V.; Petrichkovich V. M. |
| spellingShingle | Zabavskii, B. V. Petrichkovich, V. M. Забавський, Б. В. Петричкович, В. М. On the stable range of matrix rings |
| title | On the stable range of matrix rings |
| title_alt | Про стабільний ранг кілець матриць |
| title_full | On the stable range of matrix rings |
| title_fullStr | On the stable range of matrix rings |
| title_full_unstemmed | On the stable range of matrix rings |
| title_short | On the stable range of matrix rings |
| title_sort | on the stable range of matrix rings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3123 |
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