On one-sided Lie nilpotent ideals of associative rings

On one-sided Lie nilpotent ideals of associative rings

We prove that a Lie nilpotent one-sided ideal of an associative ring R is contained in a Lie solvable two-sided ideal of R. An estimation of derived length of such Lie solvable ideal is obtained depending on the class of Lie nilpotency of the Lie nilpotent one-sided ideal of R. One-sided Lie nilpote...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Datum:2007
Hauptverfasser: Luchko, V.S., Petravchuk, A.P.
Format: Artikel
Sprache:English
Veröffentlicht: Інститут прикладної математики і механіки НАН України 2007
Schriftenreihe:Algebra and Discrete Mathematics
Online Zugang:https://nasplib.isofts.kiev.ua/handle/123456789/152385
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:On one-sided Lie nilpotent ideals of associative rings / V.S. Luchko, A.P. Petravchuk // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 4. — С. 102–107. — Бібліогр.: 7 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine
id nasplib_isofts_kiev_ua-123456789-152385
record_format dspace
spelling nasplib_isofts_kiev_ua-123456789-1523852025-02-23T20:08:18Z On one-sided Lie nilpotent ideals of associative rings Luchko, V.S. Petravchuk, A.P. We prove that a Lie nilpotent one-sided ideal of an associative ring R is contained in a Lie solvable two-sided ideal of R. An estimation of derived length of such Lie solvable ideal is obtained depending on the class of Lie nilpotency of the Lie nilpotent one-sided ideal of R. One-sided Lie nilpotent ideals contained in ideals generated by commutators of the form […[[r₁,r₂],…],rn₋₁],rn] are also studied.. 2007 Article On one-sided Lie nilpotent ideals of associative rings / V.S. Luchko, A.P. Petravchuk // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 4. — С. 102–107. — Бібліогр.: 7 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:16D70. https://nasplib.isofts.kiev.ua/handle/123456789/152385 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description We prove that a Lie nilpotent one-sided ideal of an associative ring R is contained in a Lie solvable two-sided ideal of R. An estimation of derived length of such Lie solvable ideal is obtained depending on the class of Lie nilpotency of the Lie nilpotent one-sided ideal of R. One-sided Lie nilpotent ideals contained in ideals generated by commutators of the form […[[r₁,r₂],…],rn₋₁],rn] are also studied..
format Article
author Luchko, V.S.
Petravchuk, A.P.
spellingShingle Luchko, V.S.
Petravchuk, A.P.
On one-sided Lie nilpotent ideals of associative rings
Algebra and Discrete Mathematics
author_facet Luchko, V.S.
Petravchuk, A.P.
author_sort Luchko, V.S.
title On one-sided Lie nilpotent ideals of associative rings
title_short On one-sided Lie nilpotent ideals of associative rings
title_full On one-sided Lie nilpotent ideals of associative rings
title_fullStr On one-sided Lie nilpotent ideals of associative rings
title_full_unstemmed On one-sided Lie nilpotent ideals of associative rings
title_sort on one-sided lie nilpotent ideals of associative rings
publisher Інститут прикладної математики і механіки НАН України
publishDate 2007
url https://nasplib.isofts.kiev.ua/handle/123456789/152385
citation_txt On one-sided Lie nilpotent ideals of associative rings / V.S. Luchko, A.P. Petravchuk // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 4. — С. 102–107. — Бібліогр.: 7 назв. — англ.
series Algebra and Discrete Mathematics
work_keys_str_mv AT luchkovs ononesidedlienilpotentidealsofassociativerings
AT petravchukap ononesidedlienilpotentidealsofassociativerings
first_indexed 2025-11-24T21:46:25Z
last_indexed 2025-11-24T21:46:25Z
_version_ 1849709859099901952
fulltext Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 4. (2007). pp. 102 – 107 c© Journal “Algebra and Discrete Mathematics” On one-sided Lie nilpotent ideals of associative rings Victoriya S. Luchko and Anatoliy P. Petravchuk Dedicated to Professor V. V. Kirichenko on the occasion of his 65th birthday Abstract. We prove that a Lie nilpotent one-sided ideal of an associative ring R is contained in a Lie solvable two-sided ideal of R. An estimation of derived length of such Lie solv- able ideal is obtained depending on the class of Lie nilpotency of the Lie nilpotent one-sided ideal of R. One-sided Lie nilpotent ideals contained in ideals generated by commutators of the form [. . . [[r1, r2], . . .], rn−1], rn] are also studied. Introduction It is well-known that if I is an one-sided nilpotent ideal of an associative ring R then I is contained in a two-sided nilpotent ideal of R. Hence the following question is of interest: for which one-sided ideal I of the ring R there exists a two-sided ideal J such that J ⊇ I and J has properties like properties of I. In [5] it was noted that for an one-sided commutative ideal I of a ring R there exists a nilpotent-by-commutative two-sided ideal J of the ring R such that J ⊇ I. Note that Lie nilpotent and Lie solvable associative rings were in- vestigated by many authors (see, for example [4], [6], [7], [1]) and the structure of such rings is studied well enough. In this paper we prove that a Lie nilpotent one-sided ideal I of an associative ring R is contained in a Lie solvable two-sided ideal J of 2000 Mathematics Subject Classification: 16D70. Key words and phrases: associative ring, one-sided ideal, Lie nilpotent ideal, derived length. Jo u rn al A lg eb ra D is cr et e M at h .V. S. Luchko, A. P. Petravchuk 103 R. An estimation (rather rough) of Lie derived length of the ideal J depending on Lie nilpotency class of I is also obtained (Theorem 1). In case when the Lie nilpotent one-sided ideal I is contained in the ideal Rn of the ring R generated by all commutators of the form [. . . [[r1, r2], . . .], rn−1], rn] and the Lie derived length of I is less then n it is proved that I is contained in a nilpotent two-sided ideal of R (Theorem 2). The notations in the paper are standard. If S is a subset of an associa- tive ring R then by Annl R (S) (Annr R (S)) we denote the left (respectively right) annihilator of S in R. We also denote by R(−) the adjoint Lie ring of the associative ring R. Further, by R (−) n we denote the n-th member of the lower central series of the Lie ring R(−). Then Rn = R (−) n +R (−) n ·R = = R (−) n + R ·R (−) n is a two-sided ideal of the (associative) ring R. In par- ticular, R2 is a two-sided ideal of the ring R generated by all commutators of the form [r1, r2] = r1r2 − r2r1, r1, r2 ∈ R. If R is a Lie solvable ring (i.e. such that R(−) is a solvable Lie ring) then we denote by s(R) its Lie derived length. Analogously, by c(R) we denote Lie nilpotency class of a Lie nilpotent ring R. 1. Lie nilpotent one-sided ideals Lemma 1. Let I be an one-sided ideal of an associative ring R and Z = Z(I) be the center of I. Then there exists an ideal J in R such that J2 = 0 and [Z, R] ⊆ J . Proof. Let, for example, I be a right ideal from R. Take arbitrary ele- ments z ∈ Z, i ∈ I, r ∈ R. Then it holds z(ir)− (ir)z = 0 (since ir ∈ I). This implies the equality i(zr − rz) = 0 since z ∈ Z(I). As elements z, i, r are arbitrarily chosen then we have I[Z, R] = 0. Consider the right annihilator T = Annr R (I). It is clear that T is a two-sided ideal of the ring R (since I is a right ideal of R) what implies that [Z, R] ⊆ T . Further, for any element of the form zr − rz from [Z, R] and for any t ∈ T it holds (zr − rz)t = z(rt) − r(zt). Since rt ∈ T then z(rt) = 0. Besides, z ∈ I and therefore zt = 0 what brings the equality (zr− rz)t = 0. It means that [Z, R] · T = 0. Consider the left annihilator J = Annl T (T ). It is easy to see that J is a two-sided ideal of the ring R. From relations [Z, R] ⊆ T and [Z, R] · T = 0 we have the inclusion [Z, R] ⊆ J . It is also clear that J2 = 0. Analogously one can consider the case when I is a left ideal. Theorem 1. Let R be an associative ring and I be an one-sided ideal of R. If the subring I is Lie nilpotent then I is contained in a Lie solvable Jo u rn al A lg eb ra D is cr et e M at h .104 On one-sided Lie nilpotent ideals of associative rings two-sided ideal J of R such that s(J) ⊆ m(m+1)/2+m where m = c(I) is Lie nilpotency class of the subring I. Proof. Let for example I be a right ideal. We prove our proposition by the induction on the class of Lie nilpotency n = c(I) of the subring I. If n = 1 then I is a commutative right ideal and by Lemma 1 the ring R contains such an ideal T with zero square that it holds (I + T )/T ⊆ Z(R/T ) in the quotient ring R/T where Z(R/T ) is the center of R/T . It means that I + T is a two-sided ideal of the ring R and s(I + T ) 6 2. Clearly 2 = n+n(n+1)/2 if n = 1 and the statement of Theorem is true in case n = 1. Assume that the statement is true in case c(I) 6 n− 1 and prove it when c(I) = n. Denote by Z the center of the subring I. By Lemma 1 there exists an ideal T of R with T 2 = 0 such that [Z, R] ⊆ T . Consider the quotient ring R = R/T . Then Z = (Z + T )/T lies in the center of R and therefore Z + Z · R = Z + R · Z is a two-sided ideal of the ring R. Since Z ⊆ I = (I + T )/T the ideal Z + Z · R is Lie nilpotent of and its class of Lie nilpotency 6 m. Further, the quotient ring R/(Z + Z ·R) contains the right Lie nilpotent ideal I + (Z + Z · R)/(Z + Z · R) which is Lie nilpotent of class of Lie nilpotency 6 m − 1. By the induction assumption the last right ideal is contained in some Lie solvable ideal of the ring R/(Z + Z · R) of derived length 6 (m−1)m 2 + (m − 1). Since Z + Z ·R is Lie solvable and its derived length 6 m (even 6 [log2m] + 1 but we take a rough estimation) and we consider the quotient ring R/T where T is Lie solvable of derived length 1, one can easily see that I is contained in some Lie solvable (two-sided) ideal of derived length which does not exceed (m − 1)m 2 + (m − 1) + (m + 1) = (m + 1)m 2 + m. Analogously one can consider the case when I is right ideal. It seems to be unknown whether a sum of two Lie nilpotent associative rings is Lie solvable. So the next statement can be of interest (see also results about sums of PI-rings in [3]). Corollary 1. Let R be an associative ring which can be decomposed into a sum R = A+B of its Lie nilpotent subrings A and B. If at least one of these subrings is an one-sided ideal of R then the ring R is Lie solvable. Remark 1. The statements of Theorem 1 and its Corollary become false when we replace Lie nilpotency of one-sided ideals by Lie solvability. Really, consider full matrix ring R = M2(K) over an arbitrary field K of characteristic 6= 2. It is clear that I = {( x y 0 0 )∣ ∣ ∣ ∣ x, y ∈ K } Jo u rn al A lg eb ra D is cr et e M at h .V. S. Luchko, A. P. Petravchuk 105 is a right Lie solvable ideal of the ring R but I is not contained in any Lie solvable ideal of R since R is a non-solvable Lie ring. It is also clear that R = I + J where J = {( 0 0 z t )∣ ∣ ∣ ∣ z, t ∈ K } , i.e. the simple associative ring R is a sum of two right Lie solvable ideals. 2. On embedding of Lie nilpotent ideals in rings Lemma 2. Let R be an associative ring, A be a Lie nilpotent subring of R of Lie nilpotency class < m. If Z0 is a subring of A such that Z0 ⊆ Z(R) and Z0R ⊆ A then Zm 0 Rm = 0. Proof. Consider the two-sided ideal J = Z0 + Z0R = Z0 + RZ0 of the ring R. As J ⊆ A then [J, ..., J ] ︸ ︷︷ ︸ m = 0 by the condition c(A) < m. Further, it is easily to show that [J, J ] = [Z0 + Z0R, Z0 + Z0R] = Z2 0 [R, R]. By induction on k one can also show that [J, ..., J ] ︸ ︷︷ ︸ k = Zk 0 [R, ..., R] ︸ ︷︷ ︸ k . Then we have from the condition on J that [J, ..., J ] ︸ ︷︷ ︸ m = Zm 0 [R, ..., R] ︸ ︷︷ ︸ m = 0. This implies the equality Zm 0 Rm = Zm 0 ([R, ..., R] ︸ ︷︷ ︸ m + [R, ..., R] ︸ ︷︷ ︸ m ·R) = [J, ..., J ] ︸ ︷︷ ︸ m + [J, ..., J ] ︸ ︷︷ ︸ m ·R = 0. Lemma 3. Let R be an associative ring, I be an ideal of R. Then 1) if J is a nilpotent ideal of the subring I then J lies in a nilpotent ideal JI of the ring R such that JI ⊆ I; 2) if S = Annl I (I) (or Annr I (I)) then S is contained in a nilpotent ideal of the ring R which is contained in I. The proof of this Lemma immediately follows from Lemma 1.1.5 from [2]. Theorem 2. Let R be an associative ring and I be a Lie nilpotent one- sided ideal of R. If I ⊆ Rn and Lie nilpotency class of I is less than n then I is contained in an (associative) nilpotent ideal of R. Proof. Let for example I be a right ideal of the ring R and I ⊆ Rn. One can assume that that n > 2 because the statement of Theorem is obvious Jo u rn al A lg eb ra D is cr et e M at h .106 On one-sided Lie nilpotent ideals of associative rings in case n = 1. We fix n > 2 and prove the statement of Theorem by induction on the class of Lie nilpotency c = c(I) of the subring I. If c = 0 then I is the zero ideal and the proof is complete. Assume that the statement is true for rings R with c(I) 6 c−1 and prove it in case c(I) = c. Since I is Lie nilpotent then by Lemma 1 there exists a nilpotent ideal T of the ring R such that in the quotient ring R = R/T it holds [Z0, R] = 0 where Z0 is the center of the subring I and Z0 = (Z0 + T )/T . Then by Lemma 2 it holds the relation Zn 0 ·Rn = 0. If Zn 0 = 0 then Z0 + Z0R is a nilpotent ideal of the ring R and then the subring Z0 is contained in the nilpotent ideal J = Z0 + T of the ring R. Since in the quotient ring R/J for the right ideal (I + J)/J it holds the inequality c((I + J)/J) 6 c− 1 then by the inductive assumption (I + J)/J is contained in a nilpotent ideal S/J of the ring R/J . But then I ⊆ S where S is nilpotent ideal of the ring R. Let now Zn 0 6= 0. Then Zn 0 ⊆ Annl Rn (Rn) and since Z0 ⊆ Rn then Zn 0 is contained in a nilpotent ideal M of the ring R by Lemma 3. It is obvious that Z0 + Z0R is a nilpotent ideal of the ring R. Repeating the above considerations we see that I ⊆ S where S is a nilpotent ideal of the ring R. Corollary 2. Let R be an associative ring with condition R = [R, R]. If I is a Lie nilpotent one-sided ideal of R then there exists a nilpotent (two-sided) ideal J of the ring R such that I ⊆ J Corollary 3. Let R be a semiprime ring. Then every Lie nilpotent one- sided ideal is contained in the center Z(R) of the ring R and has trivial intersection with the ideal R2. Proof. Really since all nilpotent ideals of the ring R are zero then by Lemma 1 every Lie nilpotent one-sided ideal I is contained in Z(R). Since IR ⊆ Z then [IR, R] = I[R, R] = 0. Then from this equality we have IR2 = I([R, R] + [R, R] ·R) = 0. Denote J = I ∩R2. It is easily to show that J ⊆ Annl R2 (R2) and by Lemma 3 the intersection J lies in a nilpotent ideal of the ring R. Because the ring R is semiprime we have J = 0. References [1] B.Amberg and Ya.P.Sysak, Associative rings with metabelian adjoint group, Jour- nal of Algebra, 277 (2004), 456-473. [2] V.A.Andrunakievich and Yu.M.Ryabukhin, Radicals of algebras and structure theory, Nauka, Moscow, 1979. (in Russian). [3] B.Felzenszwalb, A.Giambruno and G.Leal, On rings which are sums of two PI- subrings: a combinatorial approach, Pacific Journal of Math., 209, no.1 (2003), 17-30. Jo u rn al A lg eb ra D is cr et e M at h .V. S. Luchko, A. P. Petravchuk 107 [4] S.A.Jennigs, On rings whose associated Lie rings are nilpotent, Bull. Amer. Math. Soc., 53 (1947), 593-597. [5] A.P.Petravchuk, On associative algebras which are sum of two almost commu- tative subalgebras, Publicationes Mathematicae (Debrecen). 53, no.1-2, (1998), 191-206. [6] R.K.Sharma and I.B.Srivastava, Lie solvable rings, Proc. Amer. Math. Soc., 94, no.1 (1985), 1-8. [7] W.Streb, Ueber Ringe mit aufloesbaren assoziirten Lie-Ringen, Rendiconti del Seminario Matematico dell’Università di Padova, 50, (1973), 127-142. Contact information Victoriya S. Luchko Kiev Taras Shevchenko University, Faculty of Mechanics and Mathematics, 64, Volodymyrska street, 01033 Kyiv, Ukraine E-Mail: vsluchko@gmail.com Anatoliy P. Pe- travchuk Kiev Taras Shevchenko University, Faculty of Mechanics and Mathematics, 64, Volodymyrska street, 01033 Kyiv, Ukraine E-Mail: aptr@univ.kiev.ua Received by the editors: 04.02.2008 and in final form 10.04.2008.

Ähnliche Einträge