On the action of derivations on nilpotent ideals of associative algebras
Let I be a nilpotent ideal of an associative algebra A over a field F and let D be a derivation of A. We prove that the ideal I + D(I) is nilpotent if char F = 0 or the nilpotency index I is less than char F = p in the case of the positive characteristic of the field F. In particular, the sum N(A) o...
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| author | Luchko, V. S. Лучко, В. С. Лучко, В. С. |
| author_facet | Luchko, V. S. Лучко, В. С. Лучко, В. С. |
| author_sort | Luchko, V. S. |
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| datestamp_date | 2020-03-18T19:44:40Z |
| description | Let I be a nilpotent ideal of an associative algebra A over a field F and let D be a derivation of A. We prove that the ideal I + D(I) is nilpotent if char F = 0 or the nilpotency index I is less than char F = p in the case of the positive characteristic of the field F. In particular, the sum N(A) of all nilpotent ideals of the algebra A is a characteristic ideal if char F = 0 or N(A) is a nilpotent ideal of index < p = char F. |
| first_indexed | 2026-03-24T02:35:45Z |
| format | Article |
| fulltext |
UDK 512.552
V. S. Luçko (Kyev. nac. un-t ym. T. Íevçenko)
O DEJSTVYY DYFFERENCYROVANYJ
NA NYL|POTENTNÁE YDEALÁ
ASSOCYATYVNÁX ALHEBR
Let I be a nilpotent ideal of an associative algebra A over a field F and let D be a derivation of A.
We prove that the ideal I + D(I) is nilpotent if char F = 0 or a nilpotent index of I is less than
char F = p in the case of positive characteristic of the field F. In particular, the sum N(A) of all
nilpotent ideals of the algebra A is a characteristic ideal provided that char F = 0 or N(A ) is a
nilpotent ideal of index < p = char F.
Nexaj I — nil\potentnyj ideal asociatyvno] alhebry A nad polem F i D — dyferencigvannq
A. Dovedeno, wo ideal I + D(I) [ nil\potentnym, qkwo char F = 0 abo indeks nil\potentnosti
I < char F = p u vypadku dodatno] xarakterystyky polq F. Zokrema, suma N (A) usix nil\po-
tentnyx idealiv alhebry A [ xarakterystyçnym idealom, qkwo char F = 0 abo N(A) — nil\po-
tentnyj ideal indeksu < p = char F.
Napomnym, çto dyfferencyrovanyem assocyatyvnoj alhebr¥ A nad polem F
naz¥vaetsq F-lynejnoe otobraΩenye D : A → A takoe, çto D ab( ) = D a b( ) +
+ aD b( ) dlq proyzvol\n¥x πlementov a, b A∈ . KaΩd¥j πlement a A∈ opre-
delqet vnutrennee dyfferencyrovanye Da : x → a x,[ ] = ax – xa, hde x A∈ .
Ydeal I alhebr¥ A naz¥vaetsq xarakterystyçeskym (yly stabyl\n¥m), esly
D I I( ) ⊆ dlq proyzvol\noho dyfferencyrovanyq D alhebr¥ A. V rabote [1]
dokazano, çto radykal Levyckoho (naybol\ßyj lokal\no nyl\potentn¥j ydeal)
assocyatyvnoho kol\ca R budet xarakterystyçeskym, esly addytyvnaq hruppa
R+
ne ymeet kruçenyq. ∏tot rezul\tat qvlqetsq analohom dlq assocyatyvn¥x
kolec yzvestnoho rezul\tata B.9Xartly [2] o xarakterystyçnosty lokal\no
nyl\potentnoho radykala alhebr¥ Ly nad polem nulevoj xarakterystyky.
Esly I — nyl\potentn¥j ydeal assocyatyvnoj alhebr¥ A y D — dyffe-
rencyrovanye A, to ydeal I + D(I), voobwe hovorq, ne qvlqetsq nyl\potentn¥m
(bolee toho, prymer, pryvedenn¥j v rabote, pokaz¥vaet, çto ydeal I + D(I) mo-
Ωet b¥t\ oçen\ dalekym po svoym svojstvam ot nyl\potentn¥x ydealov). Poπ-
tomu ynteresn¥m predstavlqetsq vopros: pry kakyx uslovyqx I + D(I) budet
nyl\potentn¥m ydealom, a takΩe kohda summa vsex nyl\potentn¥x ydealov bu-
det xarakterystyçeskym ydealom? Osnovn¥m rezul\tatom rabot¥ qvlqetsq
teorema91, v kotoroj dokazana nyl\potentnost\ ydeala I + D(I) v sluçae osnov-
noho polq xarakterystyky 0 y v sluçae char F = p > 0 y yndeksa nyl\potentnos-
ty I men\ßeho < p. Yz πtoj teorem¥ sleduet xarakterystyçnost\ summ¥ vsex
nyl\potentn¥x ydealov alhebr¥ A pry otmeçenn¥x v¥ße ohranyçenyqx
(sledstvye 1). Yzuçenyg dejstvyj dyfferencyrovanyj na ydeal¥ assocyatyv-
n¥x alhebr y kolec posvqweno mnoho rabot (sm., naprymer, [3, 4]).
Oboznaçenyq v rabote standartn¥e. Esly I — nyl\potentn¥j ydeal asso-
cyatyvnoj alhebr¥, to yndeksom nyl\potentnosty n I( ) πtoho ydeala naz¥va-
etsq takoe natural\noe çyslo n n I= ( ) , çto I n = 0 , I n− ≠1 0 . Çerez Der ( )A
oboznaçaetsq mnoΩestvo vsex dyfferencyrovanyj alhebr¥ A. Esly A —
assocyatyvnaq alhebra, to çerez N A( ) oboznaçym summu vsex nyl\potentn¥x
ydealov alhebr¥ A (πto budet lokal\no nyl\potentn¥j ydeal alhebr¥ A).
Esly D — dyfferencyrovanye assocyatyvnoj alhebr¥ A y M — F-pod-
prostranstvo yz A, to oboznaçaem dlq udobstva D M M0( ) ,= D Mk ( ) =
= D D Mk−( )1( ) dlq k ≥ 1.
© V. S. LUÇKO, 2009
1000 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 7
O DEJSTVYY DYFFERENCYROVANYJ NA NYL|POTENTNÁE YDEALÁ … 1001
V sledugwej lemme sobran¥ dlq udobstva nekotor¥e standartn¥e fakt¥ o
dyfferencyrovanyqx.
Lemma 1. Pust\ I — lev¥j (prav¥j) ydeal assocyatyvnoj alhebr¥ A ,
D ∈ Der ( )A . Tohda:
1) I + D(I) + … + D Ik ( ) — lev¥j (prav¥j) ydeal alhebr¥ A dlq k ≥ 1;
2) esly Z = Z(I) — centr ydeala I, to suwestvuet dvustoronnyj ydeal J
alhebr¥ A s J 2 0= takoj, çto Z A J,[ ] ⊆ ;
3) D xyk ( ) =
k
s
D x D y
s
k s k s
=
−∑ 0
( ) ( ) .
Dokazatel\stvo. 1. Pust\, naprymer, I — lev¥j ydeal alhebr¥ A. Oçe-
vydno, dostatoçno pokazat\, çto I + D(I) — lev¥j ydeal alhebr¥ A. Dlq pro-
yzvol\n¥x πlementov x ∈ A, i ∈ I, ymeem xD i( ) = D xi( ) – D x i( ) . Oçevydno,
D xi( ) – D x i( ) ∈ D(I) + I y potomu I + D(I) — lev¥j ydeal alhebr¥ A.
2. Sm., naprymer, [5] (lemma 1).
3. Pravylo Lejbnyca.
Lemma 2. Pust\ A — assocyatyvnaq alhebra nad proyzvol\n¥m polem, I
— ydeal yz A y D ∈ Der( )A . Tohda dlq proyzvol\n¥x natural\n¥x çysel k,
m takyx, çto k < m, spravedlyvo vklgçenye D Ik m( ) � I m k−
.
Dokazatel\stvo provedem yndukcyej po k. Esly k = 1, to pry m = 2 dlq
proyzvol\n¥x πlementov i1 , i I2 ∈ v¥polnqetsq sootnoßenye D i i( )1 2 =
= D i i( )1 2 + i D i1 2( ) y potomu πlement D i i( )1 2 ∈ I, poskol\ku I — ydeal alheb-
r¥ A. No tohda vsledstvye proyzvol\nosty v¥bora πlementov i1 , i I2 ∈ ymeet
mesto vklgçenye
D I ID I D I I I( ) ( ) ( )2 ⊆ + ⊆ .
Pust\ dokazano, çto D I m−( )1 � I m−2
pry m ≥ 2. Tohda
D I D I I D I I ID Im m m m( ) ( )= ⋅( ) ⊆ + ( )− − −1 1 1
,
y poπtomu ymeet mesto vklgçenye
D I I I I Im m m m( ) ⊆ + ⋅ =− − −1 2 1
.
∏to dokaz¥vaet utverΩdenye pry k = 1 . Pust\ dokazana spravedlyvost\ vklg-
çenyq dlq k – 1, dokaΩem eho dlq k. Ymeem
D I D D I D I Ik m k m m k m k( ) ( ) ( )= ( ) ⊆ ⊆− − + −1 1
s uçetom ynduktyvnoho predpoloΩenyq.
Lemma dokazana.
Teorema 1. Pust\ A — assocyatyvnaq alhebra nad polem F, I — nyl\po-
tentn¥j ydeal yz A yndeksa nyl\potentnosty n y D ∈ Der( )A . Tohda I +
+ D(I) — nyl\potentn¥j ydeal yz A yndeksa nyl\potentnosty ≤ n2
v sle-
dugwyx sluçaqx:
1) char F = 0; 2) char F = p > 0 y n < p.
Dokazatel\stvo. PokaΩem snaçala yndukcyej po k, çto
I D I I D Ik k k+ ( )( ) ⊆ + ( ) (1)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 7
1002 V. S. LUÇKO
dlq proyzvol\noho k v sluçae char F = 0 y pry k < p v sluçae poloΩytel\noj
xarakterystyky char F = p. Dejstvytel\no, dlq proyzvol\n¥x πlementov i1 ,
i I2 ∈ ymeem
D i D i D i i D i i i D i( ) ( ) ( ) ( ) ( )1 2
2
1 2
2
1 2 1
2
2
1
2
= − −{ }
y potomu D I( )2 � I + D I2 2( ) , tak kak I — ydeal alhebr¥ A. No tohda I( +
+ D I( ))2 � I + D I2 2( ) , t.9e. vklgçenye (1) spravedlyvo pry k = 2. Pust\ doka-
zano, çto I D I k+( ) −( ) 1 � I + D Ik k− −1 1( ) , dokaΩem vklgçenye I D I k+( )( ) �
� I + D Ik k( ) . Yspol\zuq ynduktyvnoe predpoloΩenye, poluçaem
I D I I D I I D I I D Ik k k k+( ) = +( ) +( ) ⊆ + ( )(− − −( ) ( ) ( )1 1 1 )) +( )I D I( ) .
No tohda
I D I I D I D Ik k k+( ) ⊆ + ( )− −( ) ( )1 1
. (2)
Dlq proyzvol\n¥x πlementov i Ik
k
−
−∈1
1
, i I1 ∈ ymeem
D i i
k
s
D i D ik
k
s
k
k s
s
k
( ) ( ) ( )− −
−
=
=
∑1 1 1 1
0
.
V sylu lemm¥92 v¥polnqetsq sootnoßenye D i Is
k( )− ∈1 dlq s = 0, 1, … , k – 2
y potomu
D i i i
k
k
D i D i
k
k
k
k
k
k( ) ( ) ( )−
−
−= +
−
+
1 1 0
1
1 11
−D i ik
k( )1 1 (3)
dlq nekotoroho πlementa i I0 ∈ . Esly k =
k
k −
1
ne delytsq na xarakterys-
tyku polq F (a πto v¥polnqetsq v sluçaqx char F = 0 y char F = p > k), to yz (3)
lehko sleduet, çto D i D ik
k
−
−
1
1 1( ) ( ) ∈ I + D Ik k( ) , poskol\ku
k
k
D i ik
k
⋅−( )1 1 ∈
∈ I . Vsledstvye proyzvol\nosty v¥bora πlementov i Ik
k
−
−∈1
1
y i I1 ∈ polu-
çaem
D I D I I D Ik k k− − ⊆ +1 1( ) ( ) ( )
y potomu na osnovanyy (2) ymeet mesto vklgçenye I D I k+ ( )( ) � I + D Ik k( ) .
PoloΩym v poslednem sootnoßenyy k = n, hde n — yndeks nyl\potentnosty
ydeala I. Tohda I D I n+ ( )( ) � I y potomu I D I n+ ( )( )
2
= 0, t.9e. I D I+ ( ) —
nyl\potentn¥j ydeal yndeksa nyl\potentnosty ≤ n2
.
Teorema dokazana.
Yz teorem¥ 1 lehko poluçyt\ sledugwyj rezul\tat, predstavlqgwyj sa-
mostoqtel\n¥j ynteres.
Sledstvye 1. Pust\ R — assocyatyvnaq alhebra nad polem F y N R( )
— summa vsex nyl\potentn¥x ydealov alhebr¥ R. Tohda N R( ) qvlqetsq xa-
rakterystyçeskym ydealom alhebr¥ R v sledugwyx sluçaqx:
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 7
O DEJSTVYY DYFFERENCYROVANYJ NA NYL|POTENTNÁE YDEALÁ … 1003
1) char F = 0; 2) N R( ) — nyl\potentn¥j ydeal yndeksa nyl\potentnosty
< p , hde p = char F > 0.
Otmetym, çto xarakterystyçnost\ nyl\potentnoho radykala kol\ca bez kru-
çenyq dokazana v rabote [4] s yspol\zovanyem druhyx podxodov.
Sledugwyj prymer pokaz¥vaet, çto ohranyçenye na yndeks nyl\potentnos-
ty v teoreme91 y sledstvyy91 v sluçae prostoj xarakterystyky osnovnoho polq
nel\zq opustyt\. ∏tot prymer qvlqetsq analohom dlq assocyatyvn¥x alhebr
yzvestnoho prymera modulqrnoj alhebr¥ Ly s razreßym¥m radykalom, kotor¥j
ne qvlqetsq xarakterystyçeskym ydealom vsej alhebr¥ (sm. [6, s. 88]).
Prymer 1. Pust\ A — prostaq assocyatyvnaq alhebra nad polem F xarak-
terystyky p > 0, G — cyklyçeskaq hruppa porqdka p y B = F G[ ] — hruppovaq
alhebra hrupp¥ G. Tohda tenzornoe proyzvedenye assocyatyvn¥x alhebr R =
= A BF⊗ (nad polem F) budet nenyl\potentnoj alhebroj, tak kak soderΩyt
podalhebru A F⊗ 1 , kotoraq qvlqetsq prostoj. PokaΩem, çto R soderΩyt
nyl\potentn¥j ydeal N takoj, çto dlq nekotoroho D R∈Der( ) ymeem N +
+ D N( ) = R, t.9e. N + D N( ) — nenyl\potentn¥j ydeal yz R.
Dejstvytel\no, kak yzvestno, nyl\potentn¥j radykal N1 alhebr¥ B ymeet
bazys (nad F) yz πlementov g – 1, g G∈ { }\ 1 , y pry πtom B N F\ � . Fyksyruem
πlement g , g ≠ 1 . Netrudno ubedyt\sq, çto suwestvuet dyfferencyrovanye
D B∈Der( ) takoe, çto D g( ) = 1 . ProdolΩym D na assocyatyvnug alhebru
R = A BF⊗ po pravylu
D a b a D bi i i i⊗( ) = ⊗∑ ∑ ( ) .
Neposredstvenno proverqetsq, çto D R∈Der( ) .
Dalee, lehko vydet\, çto nyl\radykal alhebr¥ R = A BF⊗ sovpadaet s N =
= A NF⊗ 1 , hde N1 — nyl\radykal podalhebr¥ B. Poskol\ku D g( )− 1 = 1,
to D N A F( ) ⊇ ⊗ 1 y potomu N + D N( ) = A BF⊗ = R .
Zametym, çto yndeks nyl\potentnosty ydeala N = A NF⊗ 1 raven p.
Prymenym poluçenn¥e rezul\tat¥ k yzuçenyg dejstvyq dyfferencyrova-
nyj na odnostoronnye kommutatyvn¥e ydeal¥ assocyatyvn¥x alhebr.
Teorema 2. Pust\ R — assocyatyvnaq alhebra nad polem xarakterystyky
ne ravnoj 2 y I — kommutatyvn¥j prav¥j (lev¥j) ydeal, D R∈Der( ) . Toh-
da I + D I( ) — prav¥j (lev¥j) ydeal yz R y pry πtom podalhebra I + D I( )
soderΩyt nyl\potentn¥j ydeal T yndeksa nyl\potentnosty ≤ 4 s kommu-
tatyvnoj faktor-alhebroj I D I T+( )( ) / .
Dokazatel\stvo. Rassmotrym sluçaj levoho ydeala I. V sylu lemm¥91 al-
hebra R soderΩyt ydeal J s J 2 0= takoj, çto I R J,[ ] ⊆ . Sohlasno teore-
me91 dvustoronnyj ydeal J D J+ ( ) alhebr¥ R nyl\potenten yndeksa nyl\po-
tentnosty ≤ 4 . Dalee, na osnovanyy lemm¥91 summa I + D I( ) qvlqetsq lev¥m
ydealom alhebr¥ R. PokaΩem, çto v R v¥polnqetsq sootnoßenye I[ + D I( ) ,
R] � J D J+ ( ) . Dejstvytel\no, dlq proyzvol\n¥x πlementov i ∈ I, x R∈ yme-
em D i x( ),[ ] = D i x,[ ]( ) – i D x, ( )[ ] y tak kak
D i x D J, ( )[ ]( ) ∈ , i D x J, ( )[ ] ⊆ ,
to D i x( ),[ ] ∈ J D J+ ( ) . Sledovatel\no, D I R( ),[ ] � J D J+ ( ) . No tohda v¥-
polnqetsq takΩe sootnoßenye I D I R+[ ]( ), � J D J+ ( ) .
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 7
1004 V. S. LUÇKO
Rassmotrym teper\ pereseçenye T = I D I+( )( ) ∩ J D J+( )( ) . Oçevydno, T —
nyl\potentn¥j ydeal podalhebr¥ I + D I( ) yndeksa nyl\potentnosty ≤ 4 y
pry πtom spravedlyvo vklgçenye
I D I I D I I D I R J D J+ +[ ] ⊆ +[ ] ⊆ +( ), ( ) ( ), ( ) .
No tohda I D I I D I+ +[ ]( ), ( ) � T y potomu faktor-alhebra ( ( ))/I D I T+ kom-
mutatyvna.
Teorema dokazana.
1. Letzter G. Derivations and nil ideals // Rend. Circolo Mat. Palermo. – 1988. – 37, # 2. – P. 174 –
176.
2. Hartley B. Locally nilpotent ideals of a Lie algebra // Proc. Cambridge Phil. Soc. – 1967. – 63. –
P. 257 – 272.
3. Lanski C. Left ideals and derivations in semiprime rings // J. Algrbra. – 2004. – 277, Isuue 2. –
P. 658 – 667.
4. Jordan D. A. Noetherian Ore extensions and Jacobson rings // J. London Math. Soc. – 1975. –
Ser. 10. – P. 281 – 291.
5. Luchko V. S., Petravchuk A. P. On one-sided ideals of associative rings // Algebra and Discrete
Math. – 2007. – # 3. – P. 1 – 6.
6. DΩekobson N. Alhebr¥ Ly. – M.: Myr, 1964. – 355 s.
Poluçeno 21.04.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 7
|
| id | umjimathkievua-article-3075 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:35:45Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/62/6ffa7787cb62e92f8228c9966956b562.pdf |
| spelling | umjimathkievua-article-30752020-03-18T19:44:40Z On the action of derivations on nilpotent ideals of associative algebras О действии дифференцирований на нильпотентные идеалы ассоциативных алгебр Luchko, V. S. Лучко, В. С. Лучко, В. С. Let I be a nilpotent ideal of an associative algebra A over a field F and let D be a derivation of A. We prove that the ideal I + D(I) is nilpotent if char F = 0 or the nilpotency index I is less than char F = p in the case of the positive characteristic of the field F. In particular, the sum N(A) of all nilpotent ideals of the algebra A is a characteristic ideal if char F = 0 or N(A) is a nilpotent ideal of index < p = char F. Нехай $I$ — нільпотентний ідеал асоціативної алгебри $A$ над полем $F$ i $D$ — диференціювання $A$. Доведено, що ідеал $I + D(I)$ є нільпотентним, якщо char $F = 0$ або індекс нільпотентності $I < char F = p$ у випадку додатної характеристики поля $F$. Зокрема, сума $N (A)$ усіх нільпотентних ідеалів алгебри $A$ є характеристичним ідеалом, якщо char $F = 0$ або $N(A)$ — нільпотентний ідеал індексу $< p =$ char $F$. Institute of Mathematics, NAS of Ukraine 2009-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3075 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 7 (2009); 1000-1004 Український математичний журнал; Том 61 № 7 (2009); 1000-1004 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3075/2899 https://umj.imath.kiev.ua/index.php/umj/article/view/3075/2900 Copyright (c) 2009 Luchko V. S. |
| spellingShingle | Luchko, V. S. Лучко, В. С. Лучко, В. С. On the action of derivations on nilpotent ideals of associative algebras |
| title | On the action of derivations on nilpotent ideals of associative algebras |
| title_alt | О действии дифференцирований на нильпотентные идеалы ассоциативных алгебр |
| title_full | On the action of derivations on nilpotent ideals of associative algebras |
| title_fullStr | On the action of derivations on nilpotent ideals of associative algebras |
| title_full_unstemmed | On the action of derivations on nilpotent ideals of associative algebras |
| title_short | On the action of derivations on nilpotent ideals of associative algebras |
| title_sort | on the action of derivations on nilpotent ideals of associative algebras |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3075 |
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