Semiderivations with Power Values on Lie Ideals in Prime Rings
Let R be a prime ring, let L a noncentral Lie ideal, and let f be a nonzero semiderivation associated with an automorphism σ such that f(u) n = 0 for all u ∈ L; where n is a fixed positive integer. If either Char R > n + 1 or Char R = 0; then R satisfies s 4; the standard identity in four va...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508368447209472 |
|---|---|
| author | Huang, Shuliang Хуанг, Шулян |
| author_facet | Huang, Shuliang Хуанг, Шулян |
| author_sort | Huang, Shuliang |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:16:10Z |
| description | Let R be a prime ring, let L a noncentral Lie ideal, and let f be a nonzero semiderivation associated with an automorphism σ such that f(u) n = 0 for all u ∈ L; where n is a fixed positive integer. If either Char R > n + 1 or Char R = 0; then R satisfies s 4; the standard identity in four variables. |
| first_indexed | 2026-03-24T02:24:06Z |
| format | Article |
| fulltext |
UDC 512.5
Shuliang Huang (Chuzhou Univ., China)
SEMIDERIVATIONS WITH POWER VALUES
ON LIE IDEALS IN PRIME RINGS*
НАПIВПОХIДНI З СТЕПЕНЕВИМИ ЗНАЧЕННЯМИ
НА IДЕАЛАХ ЛI У ПРОСТИХ КIЛЬЦЯХ
Let R be a prime ring, L a noncentral Lie ideal, and f a nonzero semiderivation associated with an automorphism σ such
that f(u)n = 0 for all u ∈ L, where n is a fixed positive integer. If either CharR > n + 1 or CharR = 0, then R
satisfies s4, the standard identity in four variables.
Нехай R — просте кiльце, L — нецентральний iдеал Лi та f — ненульова напiвпохiдна, асоцiйована з автоморфiзмом
σ таким, що f(u)n = 0 для всiх u ∈ L, де n — фiксоване натуральне число. Якщо CharR > n+1 або CharR = 0,
то R задовольняє стандартну тотожнiсть s4 у чотирьох змiнних.
1. Introduction. The standard identity s4 in four variables is defined as follows:
s4 =
∑
(−1)τXτ(1)Xτ(2)Xτ(3)Xτ(4)
where (−1)τ is the sign of a permutation τ of the symmetric group of degree 4.
In all that follows, unless stated otherwise, R always denotes a prime ring, Z(R) the center of
R, Q its Martindale quotient ring. The center of Q, denoted by C, is called the extended centroid of
R (we refer the reader to [1] for these objects). It is well-known that C is a field. For any x, y ∈ R,
the symbol [x, y] stands for Lie commutator xy − yx. An additive subgroup U of R is said to be
a Lie ideal of R if [u, r] ∈ U for all u ∈ U and r ∈ R. For subsets A, B of R we let [A,B] be
the additive subgroup generated by all [a, b] with a ∈ A and b ∈ B. Recall that a ring R is prime
if for any a, b ∈ R, aRb = (0) implies a = 0 or b = 0, and is semiprime if for any a ∈ R,
aRa = (0) implies a = 0. In [2], Bergen introduced the notion of a semiderivation: an additive
mapping f : R −→ R is called a semiderivation associated with a function g : R −→ R such that
f(xy) = f(x)g(y) + xf(y) = f(x)y + g(x)f(y) and f(g(x)) = g(f(x)) hold for all x, y ∈ R. In
case g = 1R, the identity map on R, f is of course a derivation. Brešar [4] proved that the only
semiderivations of prime rings are ordinary derivations and mappings of the form f(x) = λ(x−g(x)),
where λ ∈ C and g is an endomorphism.
This paper is included in a line of investigation in the literature concerning derivations having
nilpotent values. The first result is due to Herstein [10] who proved that if R is a prime ring and d
is an inner derivation of R satisfying d(x)n = 0 (resp. d(x)n ∈ Z(R)) for all x ∈ R, where n is a
fixed integer, then d = 0 (resp. R satisfies s4). In [8], Carini and Giambruno studied the case when
d(u)n(u) = 0 for all u ∈ L, a Lie ideal of R and they proved d(L) = 0, when R is a prime ring,
CharR 6= 2 and R contains no nil right ideals, and they obtained the same conclusion when n is
fixed and R is a semiprime ring with CharR 6= 2. Later in [13], Lanski obtained the same results,
removing both the bound on the indices of nilpotence and the characteristic assumption on R. In
* This research was supported by the Natural Science Research Foundation of Anhui Provincial Education
Department (No. KJ2012B125) and also by the Anhui Province College Excellent Young Talents Fund Project
(No. 2012SQRL155;2012SQRL156) of China.
c© SHULIANG HUANG, 2013
870 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
SEMIDERIVATIONS WITH POWER VALUES ON LIE IDEALS IN PRIME RINGS 871
[14], Lee extended Herstein’s result to the case of generalized derivations. More recently, Chang [5]
proved Herstein’s above result in the setting of right generalized β-derivations. Motivated by the
above results, our purpose here is to obtain some information on the structure of a prime ring R
satisfying f(u)n = 0 on a noncentral Lie ideal L, where f is a nonzero semiderivation of R and n is
a fixed positive integer.
2. Main results.
Theorem 2.1. Let R be a prime ring, L a noncentral Lie ideal and f a nonzero semiderivation
associated with an automorphism σ such that f(u)n = 0 for all u ∈ L, where n is a fixed positive
integer. If either CharR > n + 1 or CharR = 0, then R satisfies s4, the standard identity in four
variables.
Proof. If σ = 1R, then f is a derivation, and we are done by a result of Bergen and Carini [3].
So we assume next that σ 6= 1R. In this case, it is well-known that there exists a nonzero two-sided
ideal I of R such that 0 6= [I,R] ⊆ L. In particular, [I, I] ⊆ L, hence without loss of generality
we may assume that L = [I, I] ⊆ L. In view of Brešar [4] (Theorem), f(x) = λ(x − σ(x)) for all
x ∈ R, where 0 6= λ ∈ C. We are given that (λ[x, y]− λσ[x, y])n = 0, which implies that
(σ[x, y]− [x, y])n = ([σ(x), σ(y)]− [x, y])n = 0 for all x, y ∈ Q. (2.1)
By Kharchenko’s theorem [12], we divide the proof into two cases.
Case 1. Let σ be Q-outer. Since either CharR > n+ 1 or CharR = 0, by Chuang [7] (Main
theorem), we see that ([u, v] − [x, y])n = 0 for all u, v, x, y ∈ I, in particular, letting x = 0 then
[u, v]n = 0 for all u, v ∈ I. Then by Herstein [10] (Theorem 2) R is commutative, a contradiction.
Case 2. Suppose now that σ is Q-inner, then there exists an invertible element b ∈ Q − C
such that σ(x) = b−1xb for all x ∈ R. By Chuang [6] (Theorem 2), I, R and Q satisfy the same
generalized polynomial identities (or GPIs in brief), from (2.1) we have
([b−1xb, b−1yb]− [x, y])n = 0 for all x, y ∈ Q. (2.2)
In case the center C of Q is infinite, we have ([b−1xb, b−1yb]− [x, y])n = 0 for all x, y ∈ Q
⊗
C C,
where C is the algebraic closure of C. Since both Q and Q
⊗
C C are prime and centrally closed
[9] (Theorem 2.5 and Theorem 3.5), we may replace R by Q or Q
⊗
C C according as C is finite
or infinite. Thus we may assume that R is centrally closed over C (i.e., RC = R) which is either
finite or algebraically closed and ([b−1xb, b−1yb] − [x, y])n = 0 for all x, y ∈ R. By Martindale
[15] (Theorem 3), RC (and so R) is a strongly primitive ring. In light of Jacobson’s theorem [11,
p. 75], R is isomorphic to a dense ring of linear transformations of a vector space V. Let RV be a
faithful irreducible left R-module with commuting division D = End(RV ). Since C is either finite or
algebraically closed, we know that D must coincide with C. By the density theorem, R acts densely
on VD. For any given v ∈ V, we want to show that v and bv are linearly D-dependent. If bv = 0
then v and bv are D-dependent and we are done in this case. Suppose that bv 6= 0, v and bv are
D-independent. We consider the following two cases.
Subcase 1. Assume that v, bv, b−1v are D-independent. Then by the density of R, there exist
x, y ∈ R such that
xv = bv, xbv = v, yv = bv, ybv = 0.
Application of (2.2) yields that
0 =
(
[b−1xb, b−1yb]− [x, y]
)n
v = (−2)nv 6= 0
a contradiction.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
872 SHULIANG HUANG
Subcase 2. Otherwise, v, bv, b−1v are D-dependent. Since v and bv are D-independent, then
b−1v = vd1 + bvd2 for some d1, 0 6= d2 ∈ D. By the density of R, there exist x, y ∈ R such that
xv = 0, xbv = v, yv = b−1v = vd1 + bvd2, ybv = bvd1.
In view of (2.2), we have
0 =
(
[b−1xb, b−1yb]− [x, y]
)n
v = 2nvdn2 6= 0
a contradiction. From the above we have proven that bv = vαv for all v ∈ V, where αv ∈ D
depends on v ∈ V. In fact, it is easy to check that αv is independent of the choice of v ∈ V. Indeed,
for any v, w ∈ V, by the above arguments, there exist αv, αw, αv+w ∈ D such that bv = vαv,
bw = wαw, b(v + w) = (v + w)αv+w and so vαv + wαw = b(v + w) = (v + w)αv+w. Hence
v(αv − αv+w) + w(αw − αv+w) = 0. If v and w are D-independent, then αv = αv+w = αw
and we are done. Otherwise, v and w are D-dependent, say v = λw for some λ ∈ D. Thus
vαv = bv = bλw = λbw = λwαw = vαw, that is V (αv − αw) = 0. Since V is faithful, hence
αv = αw. So we conclude that there exists δ ∈ D such that bv = vδ for all v ∈ V. We claim that
δ ∈ Z(D), the center of D. Indeed, for any β ∈ D, we have b(vβ) = (vβ)δ = v(βδ) and on the
other hand b(vβ) = (bv)β = (vδ)β = v(δβ). Therefore V (βδ− δβ) = 0 and hence βδ = δβ, which
implies that δ ∈ Z(D). So b ∈ C, a contradiction.
Theorem 2.1 is proved.
Proceeding on same lines with necessary variations, we can prove the following theorem.
Theorem 2.2. Let R be a prime ring, I a nonzero ideal and f a nonzero semiderivation
associated with an automorphism σ such that f([x, y])n = 0 for all x, y ∈ I, where n is a fixed
positive integer. If either CharR > n+ 1 or CharR = 0, then R is commutative.
The following example demonstrates that R to be prime is essential in Theorem 2.2.
Example 2.1. Let Z be the ring of all integers. Set
R =
0 a b
0 0 c
0 0 0
|a, b, c ∈ Z
and I =
0 a b
0 0 0
0 0 0
|a, b ∈ Z
.
Next, let us define a mapping f : R −→ R given by
f
0 a b
0 0 c
0 0 0
=
0 2a 0
0 0 2c
0 0 0
.
The fact
0 1 0
0 0 0
0 0 0
6= 0 implies that
0 1 0
0 0 0
0 0 0
0 a b
0 0 c
0 0 0
0 1 0
0 0 0
0 0 0
= 0,
proving R is not prime. Then, it is straightforward to check that I is a nonzero ideal of R and T is a
nonzero semiderivation of R. And it is easy to find that
(
f([x, y])
)n
= 0 for all x, y ∈ I. However
R is not commutative.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
SEMIDERIVATIONS WITH POWER VALUES ON LIE IDEALS IN PRIME RINGS 873
1. Beidar K. I., Martindale W. S., Mikhalev V. Rings with generalized identities // Monogr. and Textbooks in Pure and
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P. 209 – 213.
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P. 155 – 160.
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P. 130 – 171.
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Received 10.02.12,
after revision — 11.12.12
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
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| spelling | umjimathkievua-article-24732020-03-18T19:16:10Z Semiderivations with Power Values on Lie Ideals in Prime Rings Напівпохідні з степеневими значеннями на ідеалах Лi у простих кільцях Huang, Shuliang Хуанг, Шулян Let R be a prime ring, let L a noncentral Lie ideal, and let f be a nonzero semiderivation associated with an automorphism σ such that f(u) n = 0 for all u ∈ L; where n is a fixed positive integer. If either Char R > n + 1 or Char R = 0; then R satisfies s 4; the standard identity in four variables. Нехай $R$ — просте кільцє, $L$ — нецентральний ідеал Лі та $f$ — ненульова напівпохідна, асоційована з автоморфiзмом a таким, що $f (u)^n = 0$ для всіх $u Є L$, де $n$ — фіксоване натуральне число. Якщо Char $R > n + 1$ або Char $R = 0$, то $R$ задовольняє стандартну тотожність $s_4$ у чотирьох змінних. Institute of Mathematics, NAS of Ukraine 2013-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2473 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 6 (2013); 870–873 Український математичний журнал; Том 65 № 6 (2013); 870–873 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2473/1712 https://umj.imath.kiev.ua/index.php/umj/article/view/2473/1713 Copyright (c) 2013 Huang Shuliang |
| spellingShingle | Huang, Shuliang Хуанг, Шулян Semiderivations with Power Values on Lie Ideals in Prime Rings |
| title | Semiderivations with Power Values on Lie Ideals in Prime Rings |
| title_alt | Напівпохідні з степеневими значеннями на ідеалах Лi у простих кільцях |
| title_full | Semiderivations with Power Values on Lie Ideals in Prime Rings |
| title_fullStr | Semiderivations with Power Values on Lie Ideals in Prime Rings |
| title_full_unstemmed | Semiderivations with Power Values on Lie Ideals in Prime Rings |
| title_short | Semiderivations with Power Values on Lie Ideals in Prime Rings |
| title_sort | semiderivations with power values on lie ideals in prime rings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2473 |
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