EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 2264-2275
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

D-Semiprime Rings

Maram Alosaimi1,∗, Ahmad Al Khalaf1, Rohaidah Masri2, Iman Taha1

1 Department of Mathematics and Statistic, Faculty of Sciences, Imam Mohammad Ibn Saud
Islamic University, Riyadh, Riyadh, Saudi Arabia
2 Department of Mathematics, Faculty of Sciences and Mathematics, Sultan Idris Universiti,
Tanjong Malim, Perak, Malaysia

Abstract. Let R be an associative and 2-torsion-free ring with an identity. in this work, we will
generaliz the results of differentially prime rings in [18] by applying the hypotheses in a differentially
semiprime rings. In particular, we have proved that if R is a D-semiprime ring, then either R is a
commutative ring or D is a semiprime ring.

2020 Mathematics Subject Classifications: 16W25, 16N60

Key Words and Phrases: Derivation, semiprime ring, δ-semiprime ring, δ-ideal

1. Introduction

Let R be an associative ring with an identity element. We say that R is 2-torsion-
free if for any r ∈ R and an integer n, the condition 2r = 0 holds if and only if r = 0.
Z(R) is the center of R. D is the set of all derivations in R and � is a non-empty
subset of D. An additive subgroup A is said to be a Lie ideal of R if [r, a] ∈ A, for
all r ∈ R and a ∈ A. A Lie ideal A of R is called �- ideal if δ(a) ∈ A, for all a ∈ A
and δ ∈ �. annT = {x ∈ R | xT = Tx = 0} is the annihilator of T . If a ∈ R, then
∂a(x) = [x, a] = ax − xa is an inner derivation of R induced by a ∈ R, where ∂a ∈ D.
ID = {∂a | a ∈ R} is an ideal of a ring D, see [13].

A ring R is called a �-prime (differentilly prime) if for each �-ideals A and B of R with
AB = 0, implies that A = 0 or B = 0. A ring R is said to be �-semiprime (differentilly
semiprime) if for every �-ideal I of R, the condition I2 = 0, implies that I = 0. C(R)
is the commutator ideal of R and charR is the characteristic of a ring R. By Z0(R) we
denote the ideal of R generated by its central ideals.

The properties of differentially prime rings were studied by Herstein [7, 8] and also in
his book [9]. After that, many authors have proved some results about this concept, such

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5210

Email addresses: mhalosaimi@imamu.edu.sa (M. Alosaimi),
ajalkalaf@imamu.edu.sa (A. Al Khalaf), rohaidah@fsmt.upsi.edu.my (R. Masri),
tfaith80@gmail.com (I. Taha)

https://www.ejpam.com 2264 © 2024 EJPAM All rights reserved.



M. Alosaimi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2264-2275 2265

as Bergen and Herstein [3], Hirano [11], Hongan and Trzepizur [12], Beidar and Mikhalev
[2], Chebotar and Lee [6] and could be seen in Lee and Liu [23].

Al Khalaf and others, see [18, 19, 27, 28], have demonstrated the differentially prime
rings, simple rings, differentially δ- prime rings and reverse derivation on δ- prime rings.
Furthermore, they discussed the differentially semiprime and semiprime gamma rings, see
that in [20, 21].

Many authors have investigated Lie rings of differentially semiprime rings as [14], [24]
and Jordan in [15–17] and Nowicki [25].

The commutative rings with semiprime Lie rings were studied by Passman [26] and
Bresar [4].

Finally, all other definitions and facts are standard, which were be found in [1, 13] and
also in [10].

2. Preliminaries

For any associative Lie ring R, the commutator [R,R] is a subgroup of R, which is an
additive subgroup generated by all [s, t] with s, t ∈ R.

For what we will prove, we need some lemmas.

Lemma 1. The following conditions are equivalent:

(1) R is �- semiprime ring,

(2) For any �-ideals A and B of R, the implication

AB = 0 ⇒ A ∩B = 0

is true.

(3) If a ∈ R, such that
aRδm1

1 ...δmn
n (a) = 0,

for any integers n ≥ 1,mi ≥ 0 and any derivation δi ∈ �, where i = 1, ..., n, then
a = 0.

proof. A simple modification of Proposition 2 from [22].

Lemma 2. [1] Let A be a Lie �-ideal of a �-semiprime ring R of charR ̸= 2. If [A,A] ⊆
Z(R), then A ⊆ Z(R).

Lemma 3. [1] Let R be a 2-torsion- free �-semiprime ring and A a nonzero Lie �-ideal
of R and an associative subring. Then A ⊆ Z(R) or A contains a non-central associative
�-ideal of R.

Lemma 4. [20] If R is a D-semiprime ring, Φ an ideal of D. Then

[Φ, ID] = 0 ⇔ Φ ∩ ID = 0.



M. Alosaimi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2264-2275 2266

3. Lie ideals in D-semiprime rings

Lemma 5. Let R be a 2-torsion-free �-semiprime ring, U its nonzero Lie �-ideal and an
associative subring, where a ∈ R. If

[δm1
1 · · · δmk

k (a), [a, U ]] = 0,

for any integers mi ≥ 0, k ≥ 1 and derivations δi ∈ �, where i = 1, . . . , k, then a ∈ Z(R).

proof. Let Xa = {[δm1
1 · · · δmk

k (a), x]}, x, a ∈ R, δi ∈ �,mi ≥ 0 and x, y ∈ R. From

[b, xy] = [b, x]y + x[b, y], b ∈ Xa, (1)

we get a[b, xy] = 0, then ax[b, y] = 0. Hence ayx[b, y] = 0 and yax[b, y] = 0. Thus, we
deduce that

(R[a, y]R)2 = 0, a ∈ R. (2)

In addition
0 = d(a[b, x]) = d(a)[b, x].

Multiply the identity (1) from the left by d(a), then we get d(a)x[b, y] = 0. Therefore,

0 = δ(ax[d(b), y] = δ(a)x[d(b), y],

and by the similar argument, we have

δm1
1 · · · δmk

k (a)x[δm1
1 · · · δmk

k (a), y] = 0,

for any integers k ≥ 1,mi ≥ 0 and derivations δi ∈ �, where i = 1, ..., k. As in the proof of
the condition (2), we deduce that (R[δm1

1 · · · δmk
k (a), y]R)2 = 0. Then,

I =
∞∑
k=1

∑
δi ∈ �

R[δm1
1 · · · δmk

k (a), y]R, y ∈ R

is a sum of nilpotent ideals, therefore it will be a nil ideal as well. Since I is a �- ideal, we
get I = o, hence a ∈ Z(R)

By the same way, we prove the following Lemma

Lemma 6. Let R be a 2-torsion-free �-semiprime ring, U its Lie �-ideal. If

a ∈ CR([δ
s1
1 · · · δsll (a), U ]),

for any integers si ≥ 0, l ≥ 1 and derivations δi ∈ �, where i = 1, . . . , l. Then a ∈ CR(U).



M. Alosaimi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2264-2275 2267

proof. Let u, v ∈ U , si ≥ 0, l ≥ 1 be any integers and φ, δi ∈ � be any derivation,
where i = 1, . . . , l. Since

φ(δs11 · · · δsll (a)[a, x]) = φ([a, x]δs11 · · · δsll (a)),

we have that,
δs11 · · · δsll (a) ∈ CR([δ

s1
1 · · · δsll (a), x]).

Then from

δs11 · · · δsll (a)[δ
s1
1 · · · δsll (a), uv] = [δs11 · · · δsll (a), uv]δ

s1
1 · · · δsll (a)

it holds that
[δs11 · · · δsll (a), u][δ

s1
1 · · · δsll (a), v] = 0,

what forces
[δs11 · · · δsll (a), u]t[δ

s1
1 · · · δsll (a), v] = 0,

where t ∈ R. Thus, the sum of nilpotent idal of R is �-ideal. Then a ∈ C(R).
Now, we will extend result given by [8, Theorem 3].

Proposition 1. Let R be a 2-torsion-free �-semiprime ring, W its associative �-subring
and U its Lie �-ideal. If

[W,U ] ⊆ W,

then [W,U ] = 0 or W contains a non-zero associative �-ideal of R.

proof. Let x, y, r ∈ R, t1, t ∈ U ∩W and v, w,w1, s, a, b ∈ W . Assume that

[W,U ] ̸= 0.

By Lemma 2, [U,U ] ̸= 0. Since the subring U of R generated by U satisfies that δ(U) ⊆ U ,
for all δ ∈ �, then, as in the proof of [8, Theorem 3], we can obtain that

R[a, b]RzR ⊆ UzR ⊆ W,

where z = [s, t][t, w]. Thus, the sum of nilpotent idal of R is �-ideal of R contained in W .
Otherwise

[a, b]RzR = 0,

and consequently

A =
∑

s, w ∈ W
t ∈ U ∩W

RzR

is a �-ideal such that
[a, b] ∈ annlA.

But A is non-zero and A ∩ annlA = 0, implies that

[a, b] = 0.



M. Alosaimi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2264-2275 2268

Inasmuch b = z ∈ U , we have

[z,R] ⊆ U and δm1
1 · · · δmk

k (z) ∈ U,

for any integers k ≥ 1, mi ≥ 0 and derivations δi ∈ � (i = 1, . . . , k), we conclude that

[δm1
1 · · · δmk

k (z), R] ⊆ U

what gives that
z ∈ CR([δ

m1
1 · · · δmk

k (z), R]).

By Lemma 5, z ∈ Z(R). Then

B =
∑

s, w ∈ W
t ∈ U ∩W

[s, t][t, w]R ⊆ W

is a �-ideal of R. Therefore, B = 0 and, as a consequence, z = 0. This means that

[s, t][t, w] = 0. (3)

Replace w by vw in the identity (3). Then [s, t]v[t, w] = 0 and consequently

[s, t]W [t, w] = 0. (4)

Linearize the identity (3) on t and put s = w = a; then

[a, t1][a, t] + [a, t][a, t1] = 0. (5)

Since x := [[a, t1], r] ∈ U and

2[a, t1]r[a, t1] = [x, [a, t1]] ∈ W,

we see that, using (5)
2[a, t1]R[a, t1] ⊆ W,

and, in view of the identity (4),

[s, t][a, t1]R[a, t1][w, t] = 0. (6)

In the identity (6), put s = a = w; we get

[a, t][a, t1]R[a, t][a, t1] = 0.

This means that
(R[a, t][a, t1]R)2 = 0.

Since
C =

∑
a ∈ W

t, t1 ∈ U ∩W

R[a, t][a, t1]R



M. Alosaimi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2264-2275 2269

is a �-ideal, which is a sum of nilpotent ideals, we deduce that C = 0 and so

[a, t][a, t1] = 0. (7)

We linearize the identity (7) on a to get

[a, t][b, t1] + [b, t][a, t1] = 0.

Using the previous relation in the identity (6) with w = b, we obtain

[s, t][a, t1]R[b, t1][a, t] = 0. (8)

By linearization the identity (3) for t, we have

[s, t][t1, w] + [s, t1][t, w] = 0.

In view of it, from the identity (8), by replacing b instead of s and w by a, it follows that

[s, t][a, t1]R[s, t][a, t1] = 0.

Then
D =

∑
a, s ∈ W

t, t1 ∈ U ∩W

R[s, t][a, t1]R

is a �-ideal. Then D = 0 and
[s, t][a, t1] = 0.

Denote [W, [U,W ]] by W1. Then W1 is a Lie �-ideal of R and

[s, t]W1 = 0.

Furthermore, [U,W1] ⊆ W1, [s, t]UW1 = 0 and [s, t]UW1 = 0. From the equation
R[a, b]R ⊆ U , we deduce that

[s, t]R[a, b]RW1 = 0.

Assume that p, q ∈ U ∩W , then we get

[p, q]R[p, q]RW1 = 0.

Therefore,
(R[s, t]R)3 = 0.

Then
E =

∑
p, q ∈ U ∩W

R[p, q]R

is a nil �-ideal of R, hence [p, q] = 0. As a consequence, t ∈ [U,W ] is commuting with
[U, [U,W ]]. By Lemma 3, t ∈ CR(U). Then

t ∈ CR([δ
m1
1 · · · δmk

k (t), U ]),



M. Alosaimi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2264-2275 2270

for any integers k ≥ 1, mi ≥ 0 and derivations δi ∈ �, where i = 1, . . . , k and t ∈ Z(R).
This means that [U,W ] ⊆ Z(R).

Since u ∈ U is commuting with

[δm1
1 · · · δmk

k (u), w],

for any w ∈ W , we deduce that
[U,W ] = 0,

is a contradiction.

Lemma 7. Let R be a 2-torsion-free �-semiprime ring, U its Lie �-ideal. If A ⊆ U and
satisfies that δ(A) ⊆ A for all δ ∈ � and it is an additive subgroup such that [U,A] ⊆ A
and [A,A] ⊆ Z(R), then [A,U ] = 0.

proof. Let u ∈ U and x ∈ R. If [A,A] = 0, then [a, u] ∈ A ∩ CR(a). By Lemma 5,
[A,U ] = 0. Therefore, we assume that 0 ̸= [a, b] ∈ Z(R) for some a, b ∈ A. As in the proof
of [8, Lemma 4], we can obtain that

[a, b]4 = 0.

Since [A,A] ⊆ Z(R), we deduce that

I =
∑
a,b∈A

[a, b]R

is a nil �-ideal. Then I = 0, which is a contradiction.

Corollary 1. Let R be a 2-torsion-free �-semiprime ring, U its Lie �-ideal and V satisfies
that δ(V ) ⊆ V for all δ ∈ � and it is an additive subgroup of R such that [V,U ] ⊆ V . Then
either [V,U ] = 0 or there exists a �-ideal M of R such that

0 ̸= [M,R] ⊆ V

(in particular, in the second case, V contains a non-zero Lie �-ideal of R).

proof. Clearly that A = [V,U ] ⊆ V ∩ U , where δ(A) ⊆ A for all δ ∈ � and [A,U ] ⊆
[V,U ]. Then

T = {x ∈ R | [x,R] ⊆ U}

is a �-subring of R. Let T0 be a subring of T generated by [A,A]. Then T0 satisfies that
δ(T ) ⊆ T for all δ ∈ � . Inasmuch

[[A,A], U ] ⊆ [A,A],

we have [T0, U ] ⊆ T0. By Lemma 3, [T0, U ] = 0 or T0 contains a non-zero �-ideal of R.
a) If [T0, U ] = 0, then using the fact that [A,A] ⊆ T0 we have

[[A,A], U ] = 0.



M. Alosaimi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2264-2275 2271

Since [A,A] satisfies δ([A,A]) ⊆ [A,A] for all δ ∈ �, we conclude that for a ∈ [A,A] we
have

[δm1
1 · · · δmk

k (a), [a,R]] = 0.

By Lemma 5, a will be from Z(R). This means that [A,A] ⊆ Z(R). By Lemma 7,
[A,U ] = 0. Hence by Lemma 3 A ⊆ Z(R) and [v, u] ∈ A for any v ∈ V and u ∈ U . Then

[δm1
1 · · · δmk

k (v), [v, u]] = 0

for any integers k ≥ 1, mi ≥ 0 and derivations δi ∈ � where i = 1, . . . , k, and by Lemma 5,

v ∈ CR(U)

what forces that
[V,U ] = 0.

b) Assume that T0 contains a non-zero �-ideal M of R. Then [M,R] ̸= 0 or [M,R] = 0.
In the last case

MC(R) = 0

and MT0 = 0. As a consequence, M2 = 0. By the �-semiprimeness of R, M = 0, which is
a contradiction.

Now we extended [5, Theorem 1] in the next proposition

Proposition 2. Let R be a 2-torsion-free δ-semiprime ring, U its δ-ideal, where 0 ̸= δ ∈
D. If δ2(U) = 0, then δ(U) ⊆ Z(R).

proof. Let a, b, u, v ∈ U and x, r ∈ R. From

0 = δ2([u, v]) = 2[δ(u), δ(v)],

we deduced U is commutative. Since

u[u, r] = u(ur − ru) = u(ur)− (ur)u = [u, ur] ∈ U

it follows that
0 = δ2(u[u, r]) = 2δ(u)δ([u, r])

and therefore,
δ(u)δ([u, r]) = 0.

Multiplying
[δ(u), rx] = [δ(u), r]x+ r[δ(u), x],

by δ(u) on left we get
δ(u)r[δ(u), x] = 0.

Since
δ(u)xr[δ(u), x] = 0, xδ(u)r[δ(u), x] = 0,



M. Alosaimi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2264-2275 2272

we obtain that
[δ(u), x]R[δ(u), x] = 0.

This means that
Iux = R[δ(u), x]R

is a nilpotent ideal. Inasmuch

I =
∑

u∈U,x∈R
Iux

is a nil δ-ideal, hence we conclude that I = 0. This means that δ(U) ⊆ Z(R).
Now we will investigate the inverse problem and prove the main result.

Theorem 1. Let R be a 2-torsion-free ring. If R is a D-semiprime ring then one of the
following holds:

(1) R is a commutative ring,

(2) D is a semiprime ring.

proof. Assume that R is not commutative, then C(R) ̸= 0. By the D-semiprimeness
of R, C(R)2 ̸= 0. Suppose that B is a non-zero ideal of D, where [B,B] = 0. Let
J = B ∩ ID and x, y, r, t ∈ R.

(a) If J = 0, then, for any d ∈ B,

∂d(x) = [d, ∂x] = 0

that is d(x) ∈ Z(R). Then, for any z ∈ CR(x), we obtain that

d([x, y]) = [d(x), y] + [x, d(y)] = 0,

d(z)[x, y] = d(z[x, y]) = d([x, zy]) = 0 = d([x, y]z) = [x, y]d(z)

and
rd(z)t = rtd(z) + r[d(z), t] = rtd(z).

Assume that x /∈ Z(R). The ideal Ax generated by all d(z), where d ∈ D and z ∈
CR(x), is a D-ideal of R. If Ax ̸= 0, then, using the non commutativity of a ring R
and the definition of the annihilator, we see that annAx is a non-zero D-ideal, which is
a contradiction. Hence, d(CR(x)) = 0. If d(Z(R)) = 0, then d(R) = 0 and so d = 0.
Therefore, we assume that d(Z(R)) ̸= 0. If a ∈ Z(R), then aCR(x) ⊆ CR(x) and then

d(CR(x)a) = CR(x)d(a) = 0 = d(aCR(x)) = d(a)CR(x)

and consequently
d(R) ⊆ annCR(x).



M. Alosaimi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2264-2275 2273

In view of Lemma 4, d(R) ⊆ Z(R) for d ∈ B. Let DB(R) by the ideal of R generated
by all d(R), where d ∈ B. Then u ∈ CR(u) ⊆ annDB(R) for any u ∈ DB(R) and so
u ∈ DB(R)∩annDB(R) = 0. This means that DB(R) = 0, which leads to a contradiction.

(b) Assume that J ̸= 0. Then I = {t ∈ R | ∂t ∈ J} is a non-zero D-ideal of R and

∂[t1,t2] = [∂t1 , ∂t2 ] ∈ [J, J ] = 0

for any ti ∈ J and, as a consequence,

[I, I] ⊆ Z(R). (9)

Let
T (I) = {w ∈ R | [w,R] ⊆ I}.

Then I ⊆ T (I) and T (B) is an associative subring and a Lie ideal of R (see [7, Lemma 3]).
Since

[I, T (I)] ⊆ I,

we deduce that
0 = ∂t1([t1, t

2
2]) = 2∂t1(t2)

2.

From this and the condition in the equation (9) it holds that

∂t1(t2) ∈ P(R) ∩ Z(R).

Then ∑
t1,t2∈I

[t1, t2]R

is a nil D-ideal of R, which is a contradiction.

4. Conclusion

Through this work, firstly, we found some properties of Lie ideals in D-semiprime ring,
then we demonstrated when a commutator of a composite derivation for an element and
Lie ideal in a D-semiprime ring equal to zero, implies the element belongs to center of
this ring. Also, we investigated the relationship between an element of Lie �- ideal and
the center (the commutator ideal) of a ring. After that, we showed that for any an ideal
contained in Lie �-ideal of a �-semiprime ring, their commutator must be contained in
the ideal itself. Furthermore, we related between the commutator of �- ideal and every
associative subgroup of it under specific conditions. In addition, if any δ-ideal of a δ-
semiprime ring satisfies δ2(U) = 0, this gives δ(U) is contained in the center of ring.
Finally, we proved that every a 2-torsion-free D-semiprime with an identity ring is either
commutative or a D is a semiprime ring.



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