1. Introduction
In the study of associative rings, the structure and properties of ideals play a crucial role. Prime and semiprime ideals, in particular, are significant due to their ability to characterize the ring’s behavior.
Let be an associative ring with center Z. A prime ideal in an associative ring is a proper ideal such that for any , if , then either or . A ring is called a prime ring if the zero ideal is a prime ideal, meaning that for any , implies either or .
A semiprime ideal is defined as a proper ideal where for any , if , then . A ring is termed a semiprime ring if the zero ideal is a semiprime ideal, which indicates that the ideal generated by any element squared lies in the ideal if the element itself does.
An additive mapping
on
is a derivation if
for all
. The first study on derivations of prime rings was published in 1957 by Posner [
1]. Several mathematicians have since generalized the notion of derivation in various ways, leading to the introduction of different kinds of derivations in the literature.
One such generalization is the concept of semiderivations, introduced by Bergen [
2]. An additive mapping
is a semiderivation if there exists a function
, such that
- (i)
and
- (ii)
for all . If is the identity map, semiderivations reduce to derivations, making semiderivations a broader concept.
In 1991, Bresar [
3] introduced the notion of generalized derivations: an additive mapping
on
is called a generalized derivation associated with a derivation
if
for all
. Similarly, Daif [
4] introduced multiplicative derivations by removing the additive condition, defining
as a multiplicative derivation if
for all
. Daif and Tammam El-Sayiad [
5] further extended this to multiplicative generalized derivations.
Let S be a nonempty subset of . A mapping from to is called centralizing on S if for all , and commuting on S if for all . This concept has been generalized to -commuting maps: is a -commuting map on S if for all and some .
Daif and Bell in 1992 [
6] showed that an ideal
I of a semiprime ring
is contained in the center of
if any of the following conditions are satisfied for all
:
In particular, if
, then
is commutative. This result has been extended by many authors, who have also obtained commutativity results for prime or semiprime rings with derivations satisfying certain polynomial identities (see, e.g., [
7,
8,
9,
10,
11,
12]).
The commutativity of prime and semiprime rings admitting derivations remains an active area of research. Recent approaches involve examining commutativity conditions in quotient rings
rather than assuming the ring is prime (see, e.g., [
13,
14,
15,
16]).
The principal aim of this study is to explore the definition of semiderivation given by Bergen. Inspired by the notion of multiplicative generalized derivation [
17], we introduce the concept of a multiplicative generalized semiderivation. A mapping
on
is a multiplicative generalized semiderivation if there exists a multiplicative semiderivation
associated with a map
on
, such that
- (i)
and
- (ii)
for all .
This study aims to investigate identities involving multiplicative generalized semiderivations in semiprime ideals. Since every prime ideal is semiprime, but the converse is not true, examining identities involving multiplicative generalized semiderivations in semiprime ideals is appropriate. We will prove Daif’s theorem and discuss some functional identities, extending and unifying several known results.
2. Main Results
Throughout this paper, will be an arbitrary ring, will be a semiprime ideal of , will be a multiplicative generalized semiderivation associated with a map of , and is an epimorphism on . For any as usual, and will show the associated Lie and Jordan product, respectively. The following expressions will make our work easier for us:
Theorem 1. Let be a ring admitting a semiprime ideal Y and a multiplicative generalized semiderivation ψ associated with a map θ, and γ be an epimorphism on . If any one of the following conditions is satisfied for all , then for all
- (i)
;
- (ii)
;
- (iii)
;
- (iv)
;
- (v)
;
- (vi)
.
Proof. We are assuming that
Substituting
in place of
in (
1), we obtain
This implies that
Using our hypothesis, we have
Replacing
by
in the last expression, we obtain
Since
is an epimorphism of
and using (
2), we can write this equation as
Again replacing
with
in (
3), we obtain
By right-multiplying (
3) by
we obtain
Subtracting (
4) from (
5), we arrive at
Since
is an epimorphism of
we have
Writing
instead of
t in the last equation, we obtain
Based on the semiprimeness of
we arrive at
On the other hand, by using
as a multiplicative generalized semiderivation of
and this equation, we obtain
and so
This completes the proof.
Assume that
Writing
instead of
in (
8), we have
Since
is a multiplicative generalized semiderivation of
we obtain
Using (
8), we obtain
If we put
instead of
in the last equation, we have
Expanding this equation and using (
9), we arrive at
Furthermore, the proof follows directly from Theorem 1, after Equation (
3). By applying the same technique, we obtain that
for all
Assume that . According to the hypothesis, for all We can easily obtain . Based on the semiprimeness of we arrive at for all and so a contradiction.
Now, we assume that
. We also assume that
Replacing
with
in (
10), we obtain
which can be expanded as
This can be written as
Using the hypothesis, this equation reduces to
This expression is identical to Equation (
2) in the proof of Theorem 1. By employing similar expressions within the proof of Theorem 1, we can obtain the desired result.
Assume that
. If we apply similar operations to those applied in Theorem 1 (iii), a contradiction is obtained. Alternatively, assume that
. Based on the hypothesis, we then have
Replacing
with
in the above equation and using it accordingly, we can easily obtain
By applying the same techniques used after Equation (
2) in the proof of Theorem 1, we obtain the required result.
Assume that
. If we perform the same operations as in Theorem 1 (iii), a contradiction arises. Thus, assume now that
. Based on the hypothesis, we then have
Replacing
with
in (
12) and using this equation accordingly, we find that
This expression is identical to Equation (
9) in the proof of Theorem 1. By using expressions similar to those used in the proof of Theorem 1, we can obtain the desired result.
Assume that
. If similar operations are performed in Theorem 1 (iii), a contradiction is reached. Now assume that
. Based the hypothesis, we then have
Replacing
with
in (
13) and using this equation, we arrive at the following conclusion
Again using the same lines as those used in the proof of Theorem 1, we conclude that
for all
□
Theorem 2. Let be a ring admitting a semiprime ideal Y and ψ be a multiplicative generalized semiderivation associated with a map θ, with γ being an epimorphism on . If any one of the following conditions is satisfied for all , then for all
- (i)
;
- (ii)
;
- (iii)
;
- (iv)
.
Proof. Assume that
. Based on the hypothesis,
for all
This expression is identical to the one in Theorem 1
. We now assume that
. Using our hypothesis, we obtain
Taking
instead of
within (
14), we have
Since
is a multiplicative generalized semiderivation of
we obtain
which can be written as
Using the hypothesis, we arrive at
Since
is an epimorphism of
we have
and so
Due to the semiprimeness of
we arrive at
On the other hand, we know that
according to our hypothesis, and so
for all
Using the
in the last expression, we understand that
and so
The semiprimeness of
forces us to conclude that
Taking
instead of
in (
16) and subtracting the relations (
16) and (
17), we find that
and so
This completes the proof.
Assume that
. Based on the hypothesis,
for all
. This expression is the same as the expression in Theorem 1 (iii). We now assume that
. Using our hypothesis, we obtain
and
Subtracting these two equations, we have
and so
Writing
instead of
in this equation and using it accordingly, we find that
Replacing
with
in (
18) and using (
18), we obtain
and so
Similarly, (
18) reveals that
Subtracting (
19) from (
20), we find that
and so
Due to the semiprimeness of
, we obtain
Replacing
with
in this equation and using it accordingly, we obtain
and so
Again using the semiprimeness of
, we obtain the required result.
Assume that
Define the map
for all
is a multiplicative generalized semiderivation associated with a nonzero map
of
Based on the hypothesis, we have
for all
Hence, the conclusion is obtained via Theorem 2
.
Using our hypothesis, we obtain
Define the map
for all
is a multiplicative generalized semiderivation associated with a nonzero map
of
, such that
for all
. Using Theorem 2 (ii), we find that
for all
. □
Theorem 3. Let be a ring admitting a semiprime ideal Y and ψ be a multiplicative generalized semiderivation associated with a map θ, with γ being an epimorphism on . If any one of the following conditions is satisfied for all , then for all
- (i)
;
- (ii)
;
- (iii)
;
- (iv)
.
Proof. We assume that
Writing
instead of
in (
21), we obtain
Using the definition of
, we have
Based on the hypothesis, we find that
This expression is exactly the same as Equation (
2) in the proof of Theorem 1. By using expressions similar to those used in the proof of Theorem 1, we obtain the desired result.
By using our hypothesis, we obtain
Replacing
with
in (
22) and using this equation, we find that
This expression is exactly the same as Equation (
2) in the proof of Theorem 1. By using expressions similar to those used in that proof, we obtain the desired result.
Using our hypothesis, we obtain
Replacing
with
in (
23), we obtain
Since
is a multiplicative generalized semiderivation of
we obtain
Based on the hypothesis, we find that
This expression is exactly the same as Equation (
9) in the proof of Theorem 1. By using expressions similar to those used in that proof, we obtain the desired result.
Using our hypothesis, we obtain
Replacing
with
in (
24) and using (
24), we obtain
Using the same reasoning as applied after Equation (
9) in the proof of Theorem 1, we arrive at the desired result. □
Theorem 4. Let be a ring admitting a semiprime ideal Y and ψ be a multiplicative generalized semiderivation associated with a map θ, with γ being an epimorphism on If any of the following conditions are satisfied for all , then for all
- (i)
;
- (ii)
;
- (iii)
;
- (iv)
.
Proof. Based on our hypothesis
Taking
instead of
in (
25), we obtain
and so
which can be expanded as
Using (
25), we find that
Equation (
2) yields the same expression. Similar expressions can be used in the proof of Theorem 1. Thus, the proof is complete.
Let us assume that
Writing
instead of
in (
27) and using (
27), we reveal that
Equation (
2) yields the same expression. Similar expressions can be used in the proof of Theorem 1. Therefore, the proof is complete.
We assume that
Taking
instead of
in (
28) and using this, we find that
By applying the same arguments used after Equation (
9) in the proof of Theorem 1, we obtain the required result.
Let us assume that
Replacing
with
in (
30) and using (
30), we find that
By using the same arguments as those following Equation (
9) in the proof of Theorem 1, we obtain the required result. □
Theorem 5. Let be a 2-torsion-free ring with Y being a semiprime ideal of . Suppose that admits a multiplicative generalized semiderivation ψ associated with a map If any of the following conditions are satisfied for all , then θ is a -commuting map on .
- (i)
;
- (ii)
.
Proof. Assume that
. Based on the hypothesis,
for all
This expression is the same as the one in Theorem 1
. We now assume that
. Using the hypothesis, we have
Replacing
with
in the hypothesis and using the hypothesis, we find that
Again using the hypothesis, we have
Since
is an epimorphism of
we have
Replacing
with
,
, we have
Applying (
32), we have
Since
is an epimorphism of
we have
for all
This implies that
for all
and so
for all
Hence, we obtain
Assume that
. Based on the hypothesis,
for all
This expression is the same as the expression in Theorem 1
. We now assume that
. Using the hypothesis, we have
Replacing
with
in (
33), we obtain
Using the hypothesis, we obtain
Since
is an epimorphism of
we have
Left-multiplying this equation by
for
and using it accordingly, we obtain
Based on the hypothesis, we can express the last equation as follows:
Replacing
with
we have
Taking
instead of
we obtain
Since
is a semiprime ideal of
we conclude that
Right-multiplying by
in (
34), we obtain
Left-multiplying this equation by
, we obtain
Since
is a semiprime ideal of
we conclude that
Subtracting (
35) from (
36), we find that
This completes the proof. □
Theorem 6. Let be a -torsion-free ring with Y being a semiprime ideal of . Suppose that admits a multiplicative generalized semiderivation ψ associated with a nonzero multiplicative semiderivation If any of the following conditions are satisfied for all , then θ is a -commuting map on
- (i)
;
- (ii)
.
Proof. We obtain
Replacing
with
in (
37), we have
Using (
37), we see that
Substituting
for
in the last expression, we have
Since
is an epimorphism of
we have
Replacing
with
we obtain
Using the hypothesis and
, we have
and so
Writing
instead of
in (
39) and using (
39), we find that
Replacing
with
in (
40), we find that
Multiplying (
40) on the left by
we have
Subtracting (
41) from (
42), we determine that
that is,
Since
is a semiprime ideal of
we conclude that
and so
is a
-commuting map on
We have
If we write
instead of
in (
43), then, since
is 2-torsion-free, we find that
Using (
43), we obtain
Replacing
with
in (
44), we have
and so
Since
is an epimorphism of
, we can use the same arguments as those following Equation (
38) to show that
This completes the proof. □