Remarks on Certain Identities with Derivations on Semiprime Rings
Let $n$ be a fixed positive integer, let $R$ be a $(2n)!$ -torsion-free semiprime ring, let $\alpha$ be an automorphism or an anti-automorphism of $R$, and let $D_1 , D_2 : R → R$ be derivations. We prove the following result: If $(D_1^2 (x) + D_2(x))^n ∘ α(x)^n = 0 $ holds for all $x Є R$, then $...
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| author | Baydar, N. Fošner, A. Strašek, R. Байдар, Н. Фоснер, А. Страшек, Р. |
| author_facet | Baydar, N. Fošner, A. Strašek, R. Байдар, Н. Фоснер, А. Страшек, Р. |
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| description | Let $n$ be a fixed positive integer, let $R$ be a $(2n)!$ -torsion-free semiprime ring, let $\alpha$ be an automorphism or an anti-automorphism of $R$, and let $D_1 , D_2 : R → R$ be derivations. We prove the following result: If $(D_1^2 (x) + D_2(x))^n ∘ α(x)^n = 0 $ holds for all $x Є R$, then $D_1 = D_2 = 0$. The same is true if $R$ is a 2-torsion free semiprime ring and F(x) ° β(x) = 0 for all x ∈ R, where $F(x) = (D_1^2 (x) + D_2(x)) ∘ α(x),\; x ∈ R$, and $β$ is any automorphism or antiautomorphism on $R$. |
| first_indexed | 2026-03-24T02:21:14Z |
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UDC 512.5
A. Fošner (Univ. Primorska, Slovenia),
N. Baydar (Ege Univ., Izmir, Turkey),
R. Strašek (Univ. Primorska, Slovenia)
REMARKS ON CERTAIN IDENTITIES
WITH DERIVATIONS ON SEMIPRIME RINGS
ПРО ДЕЯКI ТОТОЖНОСТI ДЛЯ ПОХIДНИХ НА НАПIВПРОСТИХ КIЛЬЦЯХ
Let n be a fixed positive integer, R a (2n)!-torsion free semiprime ring, α an automorphism or an anti-automorphism of R,
and D1, D2 : R → R derivations. We prove the following result: If (D2
1(x) +D2(x)) ◦ α(x)n = 0 holds for all x ∈ R,
then D1 = D2 = 0. The same is true if R is a 2-torsion free semiprime ring and F (x) ◦ β(x) = 0 for all x ∈ R, where
F (x) = (D1
2(x) +D2(x)) ◦ α(x), x ∈ R, and β is any automorphism or anti-automorphism on R.
Припустимо, що n — фiксоване натуральне число, R — (2n)! напiвпросте кiльце, вiльне вiд кручення, α — авто-
морфiзм або антиавтоморфiзм на R, а D1, D2 : R → R — похiднi. Доведено наcтупний результат: якщо (D2
1(x) +
+D2(x))◦α(x)n = 0 виконується для всiх x ∈ R, тоD1 = D2 = 0. Аналогiчне твердження справджується, якщо R
— 2-напiвпросте кiльце, вiльне вiд кручення, i F (x) ◦β(x) = 0 для всiх x ∈ R, де F (x) = (D1
2(x)+D2(x)) ◦α(x),
x ∈ R, i β — довiльний автоморфiзм або антиавтоморфiзм на R.
1. Introduction. The aim of this paper is to generalize the results obtained in [9]. Let us first fix
some notation. Throughout the paper, R will represent an associative ring with a center Z(R). Let
n > 1 be an integer. We say that a ring R is n-torsion free if nx = 0, x ∈ R, implies x = 0. As
usual, the Lie product of elements x, y ∈ R will be denoted by [x, y] (i. e., [x, y] = xy − yx) and the
Jordan product of elements x, y ∈ R will be denoted by x ◦ y (i.e., x ◦ y = xy + yx). Recall that a
ring R is prime if aRb = {0}, a, b ∈ R, implies that either a = 0 or b = 0, and it is semiprime if
aRa = {0}, a ∈ R, implies a = 0.
An additive mapping f : R → R is called centralizing on R if [f(x), x] ∈ Z(R) holds for all
x ∈ R. In a special case, when [f(x), x] = 0 for all x ∈ R, the mapping f is said to be commuting
on R. Furthermore, an additive mapping f : R → R is skew-centralizing on R if f(x) ◦ x ∈ Z(R)
for all x ∈ R, and it is called skew-commuting on R if f(x) ◦ x = 0 is fulfilled for all x ∈ R. We
say that an additive mapping D : R→ R is a derivation on R if D(xy) = D(x)y+ xD(y) holds for
all x, y ∈ R. A classical result of Posner [12] (Posner’s second theorem) states that the existence of a
nonzero centralizing derivation on a prime ring forces the ring to be commutative. On the other hand,
Posner’s second theorem in general cannot be proved for semiprime rings as shows the following
example. Let R1 and R2 be prime rings with R1 commutative and set R = R1⊕R2. Further, let D1 :
R1 → R1 be a nonzero derivation. Then a mapping D : R→ R given by D((r1, r2)) = (D1(r1), 0)
is a nonzero commuting derivation. It is also easy to show that every commuting derivation on a
semiprime ring R maps R into Z(R) (see, for example, the end of the proof of Theorem 2.1 in [13]).
In the present paper we continue the series of papers concerning arbitrary additive maps of
prime and semiprime rings satisfying certain identities (see [1 – 5, 9] and the references therein). In
particular, we generalize the main results obtained in [9].
2. The results. Before stating our main theorems, let us write some known facts which we will
need in the sequel. So, let R be a 2-torsion free semiprime ring and f : R→ R an additive mapping
such that [f(x), x2] = 0 holds for all x ∈ R. Then f must be commuting on R. This result was proved
c© A. FOŠNER, N. BAYDAR, R. STRAŠEK, 2014
1436 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
REMARKS ON CERTAIN IDENTITIES WITH DERIVATIONS ON SEMIPRIME RINGS 1437
by Vukman and the second named author in [9]. Moreover, the same conclusion is true, if f satisfies
[f(x), xn] = 0, x ∈ R, where n is a fixed positive integer and R is a n!-torsion free semiprime ring
(see [8], Theorem 2). Now, let α be an automorphism of R and suppose that an additive mapping f :
R→ R satisfies the relation
[f(x), α(x)n] = 0 (1)
for all x ∈ R. This means that
f(x)α(x)n − α(x)nf(x) = 0
for all x ∈ R. Since α is an automorphism of R, we have
α−1(f(x))xn − xnα−1(f(x)) = [α−1(f(x)), xn] = 0.
Moreover, if α is an anti-automorphism of R such that (1) holds, then
xnα−1(f(x))− α−1(f(x))xn = −[α−1(f(x)), xn] = 0.
Thus, using Theorem 2 in [8], we have the next result.
Proposition 1. Let n be a fixed positive integer, R a n!-torsion free semiprime ring, and α an
automorphism or an anti-automorphism of R. Suppose that an additive mapping f : R→ R satisfies
the relation (1) for all x ∈ R. Then [f(x), α(x)] = 0 holds for all x ∈ R.
Next, let us take the Jordan product instead of the Lie product in (1) and observe the relation
f(x) ◦ α(x)n ∈ Z(R), (2)
where f is an additive map on a (2n)!-torsion free semiprime ring R and α an automorphism or an
anti-automorphism of R. Then we obtain
[f(x) ◦ α(x)n, y] = 0
for all y ∈ R. Replacing y by α(x)n, we get
0 = [f(x) ◦ α(x)n, α(x)n] = [f(x), α(x)2n].
Using Proposition 1, we have the next result which generalizes Theorem 3 in [8].
Proposition 2. Let n be a fixed positive integer, R a (2n)!-torsion free semiprime ring, and α an
automorphism or an anti-automorphism of R. Suppose that an additive mapping f : R→ R satisfies
the relation (2) for all x ∈ R. Then [f(x), α(x)] = 0 holds for all x ∈ R.
In particular, we will use the following corollary of Proposition 2.
Corollary 1. Let n be a fixed positive integer, R a (2n)!-torsion free semiprime ring, and α an
automorphism or an anti-automorphism of R. Suppose that an additive mapping f : R→ R satisfies
f(x) ◦ α(x)n = 0
for all x ∈ R. Then [f(x), α(x)] = 0 holds for all x ∈ R.
Posner’s first theorem [12] states that the composition of two nonzero derivations on a 2-torsion
free prime ring cannot be a derivation. On the other hand, this conclusion is not true in the case
of semiprime rings (see, for example, [6]). However, Herstein [10] (Lemma 1.1.9) showed that if
R is a 2-torsion free semiprime ring and D1, D2 : R → R derivations such that D2
1(x) = D2(x)
holds for all x ∈ R, then D1 = D2 = 0. The same is true if D1 and D2 satisfy the relation
(D2
1(x) +D2(x)) ◦ x2 = 0 for all x ∈ R (see [9]). These results motivated us to prove the following
theorem which generalizes Theorem 8 in [9].
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
1438 A. FOŠNER, N. BAYDAR, R. STRAŠEK
Theorem 1. Let n be a fixed positive integer, R a (2n)!-torsion free semiprime ring, α an
automorphism or an anti-automorphism of R, and D1, D2 : R→ R derivations. Suppose that
(D2
1(x) +D2(x)) ◦ α(x)n = 0
holds for all x ∈ R. Then D1 = D2 = 0.
In the following, we shall use the fact that any semiprime ring R and its maximal right ring of
quotients Q satisfy the same differential identities which is very useful since Q contains the identity
element (see [11], Theorem 3). For the explanation of differential identities we refer the reader to [7].
Proof of Theorem 1. By Theorem 3 in [11], we have
D(x) ◦ α(x)n = 0 (3)
for all x ∈ Q, where D(x) stands for D2
1(x)+D2(x). Since D is additive, by Corollary 1, we obtain
[D(x), α(x)] = 0 for all x ∈ R and, again, using [11], this identity is true for all x ∈ Q.
Recall that D(1) = 0. Putting x+ 1 instead of x in (3) we get
D(x)
n∑
k=0
(
n
k
)
α(x)n−k +
n∑
k=0
(
n
k
)
α(x)n−kD(x) = 0 (4)
for all x ∈ Q. It follows from (3) and (4) that
D(x)
n∑
k=1
(
n
k
)
α(x)n−k +
n∑
k=1
(
n
k
)
α(x)n−kD(x) = 0 (5)
holds for all x ∈ Q. Again, putting x+ 1 instead of x and comparing the obtained equality with (5),
we have
D(x)
n∑
k=2
tkα(x)
n−k +
n∑
k=2
tkα(x)
n−kD(x) = 0,
where t2, . . . , tn are the appropriate positive integers. Continuing with the same procedure for
(n− 2)-times, we get
n!(D(x)α(x) + α(x)D(x)) + (n− 1)n!D(x) = 0
for every x ∈ Q. Since [D(x), α(x)] = 0, we obtain
2D(x)α(x) + (n− 1)D(x) = 0
for all x ∈ Q. Again, putting x + 1 in the last identity, we get 2D(x) = 0, x ∈ Q, and, therefore,
D = 0. Recall that in the case n = 1 we do this procedure just for one time and if n = 2 we do this
procedure for two times. In both cases we get the same conclusion, i.e., D = 0. At the end, using
Lemma 1.1.9 in [10], we get D1 = 0 and D2 = 0, as asserted.
Theorem 1 is proved.
If we take n = 2 and α = id, where id denotes the identity map on R, we have the next direct
consequence of Theorem 1.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
REMARKS ON CERTAIN IDENTITIES WITH DERIVATIONS ON SEMIPRIME RINGS 1439
Corollary 2 ([9], Theorem 8). Let R be a 2-torsion free semiprime ring and let D1, D2 : R→ R
be derivations. Suppose that
(D2
1(x) +D2(x)) ◦ x2 = 0
holds for all x ∈ R. Then D1 = D2 = 0.
Remark 1. Let us point out that in Corollary 2 we do not have to restrict ourselves to 4!-torsion
free semiprime rings, since the result holds true for 2-torsion free semiprime rings, as well. The main
idea of the proof remains the same.
We proceed with the following result which generalizes Theorem 9 in [9].
Theorem 2. LetR be a 2-torsion free semiprime ring, α an automorphism or an anti-automorphism
of R, and D1, D2 : R→ R derivations. Suppose that F : R→ R is a mapping defined by
F (x) = (D1
2(x) +D2(x)) ◦ α(x), x ∈ R.
If F (x) ◦ β(x) = 0 holds for all x ∈ R and some automorphism or anti-automorphism β of R, then
D1 = D2 = 0.
Proof. By the assumption, we have(
D(x)α(x) + α(x)D(x)
)
◦ β(x) = 0
for all x ∈ R, where D(x) = D1
2(x) +D2(x). This means that(
D(x)α(x) + α(x)D(x)
)
β(x) + β(x)
(
D(x)α(x) + α(x)D(x)
)
= 0
for all x ∈ R. According to Theorem 3 in [11], the above identity holds for all x ∈ Q. Replacing x
by x+ 1, we obtain
0 =
(
D(x)α(x) + α(x)D(x)
)
β(x) + β(x)
(
D(x)α(x) + α(x)D(x)
)
+
+2
(
D(x)α(x) + α(x)D(x)
)
+ 2
(
D(x)β(x) + β(x)D(x)
)
+ 4D(x)
for all x ∈ Q. Combining the last two relations, it follows that
D(x)α(x) + α(x)D(x) +D(x)β(x) + β(x)D(x) + 2D(x) = 0 (6)
for all x ∈ Q. Again, putting x + 1 instead of x in the above identity and comparing so obtained
equality with the relation (6), we get 4D(x) = 0 for all x ∈ Q. This yields that D(x) = 0 for all
x ∈ R and, by Lemma 1.1.9 in [10], D1 = D2 = 0.
Theorem 2 is proved.
Taking α = β = id, we have the next direct consequence of Theorem 2.
Corollary 3 ([9], Theorem 9). Let R be a 2-torsion free semiprime ring and let D1, D2 : R→ R
be derivations. Suppose that F : R→ R is a mapping defined by
F (x) = (D1
2(x) +D2(x)) ◦ x, x ∈ R.
If F is skew-commuting on R, then D1 = D2 = 0.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
1440 A. FOŠNER, N. BAYDAR, R. STRAŠEK
Remark 2. At the end, let us point out that (with the same main idea) we can prove the conclusion
of Theorem 2 even if we replace the identity F (x) ◦ β(x) = 0 with the identity F (x) ◦ β(x)n = 0,
where n is any fixed positive integer. We only have to restrict ourselves to suitable torsion free
semiprime rings. In the following, we will write just a sketch of the proof since the poof is rather
technical but the main idea remains the same.
Firstly, we know that
(D(x)α(x) + α(x)D(x)) ◦ β(x)n = 0
for all x ∈ R. This means that(
D(x)α(x) + α(x)D(x)
)
β(x)n + β(x)n
(
D(x)α(x) + α(x)D(x)
)
= 0
for all x ∈ Q, as well. Replacing x by x+ 1, we obtain
0 =
(
D(x)α(x) + α(x)D(x) + 2D(x)
) n∑
k=0
(
n
k
)
β(x)n−k+
+
n∑
k=0
(
n
k
)
β(x)n−k
(
D(x)α(x) + α(x)D(x) + 2D(x)
)
for all x ∈ Q. Combining the last two relations, it follows that
0 =
(
D(x)α(x) + α(x)D(x) + 2D(x)
) n∑
k=1
(
n
k
)
β(x)n−k+
+2
(
D(x)β(x)n + β(x)nD(x)
)
+
n∑
k=1
(
n
k
)
β(x)n−k
(
D(x)α(x) + α(x)D(x) + 2D(x)
)
.
Again, putting x + 1 instead of x in the above identity and continuing with the same procedure for
n-times, we get D(x) = 0 for all x ∈ Q. This yields that D(x) = 0 for all x ∈ R and, by Lemma
1.1.9 in [10], D1 = D2 = 0.
1. Brešar M. On a generalization of the notion of centralizing mappings // Proc. Amer. Math. Soc. – 1992. – 114. –
P. 641 – 649.
2. Brešar M. Centralizing mappings of rings // J. Algebra. – 1993. – 156. – P. 385 – 394.
3. Brešar M. Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings // Trans.
Amer. Math. Soc. – 1993. – 335. – P. 525 – 546.
4. Brešar M. On skew-commuting mappings of rings // Bull. Austral. Math. Soc. – 1993. – 47. – P. 291 – 296.
5. Brešar M., Hvala B. On additive maps of prime rings // Bull. Austral. Math. Soc. – 1995. – 51. – P. 377 – 381.
6. Brešar M., Vukman J. Orthogonal derivations and an extension of a theorem of Posner // Radovi Mat. – 1989. – 5. –
P. 237 – 246.
7. Chuang C.-L. On decomposition of derivations of prime rings // Proc. Amer. Math. Soc. – 1990. – 108. – P. 647 – 652.
8. Fošner A., Rehman N. Ur. Identities with additive mappings in semiprime rings // Bull. Korean Math. Soc. – 2014. –
51, № 1. – P. 207 – 211.
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Debrecen. – 2011. – 78. – P. 575 – 581.
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11. Lee T.-K. Semiprime rings with differential idetities // Bull. Inst. Math. Acad. Sinica. – 1992. – 20. – P. 27 – 38.
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Received 17.09.12,
after revision — 18.10.13
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
|
| id | umjimathkievua-article-2236 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:21:14Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/23/3550900e7e760bfdc9a64ced14292a23.pdf |
| spelling | umjimathkievua-article-22362019-12-05T10:26:46Z Remarks on Certain Identities with Derivations on Semiprime Rings Про деякі тотожності для похідних на напівпростих кiльцях Baydar, N. Fošner, A. Strašek, R. Байдар, Н. Фоснер, А. Страшек, Р. Let $n$ be a fixed positive integer, let $R$ be a $(2n)!$ -torsion-free semiprime ring, let $\alpha$ be an automorphism or an anti-automorphism of $R$, and let $D_1 , D_2 : R → R$ be derivations. We prove the following result: If $(D_1^2 (x) + D_2(x))^n ∘ α(x)^n = 0 $ holds for all $x Є R$, then $D_1 = D_2 = 0$. The same is true if $R$ is a 2-torsion free semiprime ring and F(x) ° β(x) = 0 for all x ∈ R, where $F(x) = (D_1^2 (x) + D_2(x)) ∘ α(x),\; x ∈ R$, and $β$ is any automorphism or antiautomorphism on $R$. Припустимо, що $n$ — фіксоване натуральне число, $R$ — $(2n)!$ напівпросте кільцє, вільнє від кручення, $\alpha$ — автоморфізм або антиавтоморфізм на $R$, а $D_1 , D_2 : R → R$ — похідні. Доведено наступний результат: якщо $(D_1^2 (x) + D_2(x))^n ∘ α(x)^n = 0 $ виконується для всіх $x Є R$, то $D_1 = D_2 = 0$. Аналогічне твердження справджується, якщо $R$ — 2-напівпросте кільце, вільне від кручення, i $F(x) ° β(x) = 0$ для всіх $x Є R$, де $F(x) = (D_1^2 (x) + D_2(x)) ∘ α(x),\; x ∈ R$, i $β$ — довільний автоморфізм або антиавтоморфізм на $R$. Institute of Mathematics, NAS of Ukraine 2014-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2236 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 10 (2014); 1436–1440 Український математичний журнал; Том 66 № 10 (2014); 1436–1440 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2236/1473 https://umj.imath.kiev.ua/index.php/umj/article/view/2236/1474 Copyright (c) 2014 Baydar N.; Fošner A.; Strašek R. |
| spellingShingle | Baydar, N. Fošner, A. Strašek, R. Байдар, Н. Фоснер, А. Страшек, Р. Remarks on Certain Identities with Derivations on Semiprime Rings |
| title | Remarks on Certain Identities with Derivations on Semiprime Rings |
| title_alt | Про деякі тотожності для похідних на напівпростих кiльцях |
| title_full | Remarks on Certain Identities with Derivations on Semiprime Rings |
| title_fullStr | Remarks on Certain Identities with Derivations on Semiprime Rings |
| title_full_unstemmed | Remarks on Certain Identities with Derivations on Semiprime Rings |
| title_short | Remarks on Certain Identities with Derivations on Semiprime Rings |
| title_sort | remarks on certain identities with derivations on semiprime rings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2236 |
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