1. Introduction
Throughout this paper
ℑ stands for an associative ring and its center is
. It is appropriate to start by recalling some well-known concepts about rings. A ring
ℑ is called a prime ring if
for each
, then either
or
. An ideal
℘ of a ring
ℑ with
is called prime if
(
) for
, which implies that either
or
. Consequently,
ℑ is a prime ring if and only if
is a prime ideal of
ℑ. We recall that a ring without non-zero divisors is a domain, and the integral domain is a commutative domain with identity. It is known that every integral domain is a prime ring and the converse needs not to be true in general. It is also known that
℘ is a prime ideal if and only if
is an integral domain. Additionally, if
℘ is an ideal in a commutative ring
ℑ, then
is commutative. It is worth mentioning prime ideal would make an interesting fertile topic to research, not only in rings, but also in algebras such as
algebras,
-algebra and Lie algebra (for more details, see refs. [
1,
2,
3,
4]). An additive map
that satisfies
for all
is called an ordinary derivation, while an additive map
which satisfies
for every
is called a generalized derivation, where
∂ is just an associated derivation map.
Suppose that
are automorphisms on
ℑ, then an additive map
is called an
-derivation if it satisfies
for any two elements
. Afterwards, this concept was expanded to a generalized
derivation as follows:
for any two elements
. Without any controversy, this concept covers the generalized derivation when
as well as the ordinary derivation when
and
, where
I the identity map on
ℑ. One of the basic problems in ring theory is to investigate the various conditions under which a ring
ℑ becomes commutative. For this purpose, there has been a great deal of effort to link the commutativity of a prime or a semiprime ring
ℑ with the existence of additive maps defined on it, such as a generalized
-derivation and an
-derivation that satisfy differential identities over the entire ring or any appropriate subset of it. For more details, the reader can refer—for example—to refs. [
5,
6,
7,
8]. As an extension of these studies, instead of proving commutativity on a prime or a semiprime ring, Almahdi et al. [
9] strengthened it without imposing any restrictions on the ring
ℑ. They proved that either
is a commutative integral domain or
, if
ℑ admits a derivation
∂ that satisfies
for any
, where
℘ is a prime ideal on
ℑ, which is generalized by the second Posner’s Theorem. Before these authors, in ref. [
10], Creedon generalized the first Posner’s Theorem in prime ideal with two iterates of derivations when a ring is restricted by a characteristic two. In this direction, studies and interests have been continued by many researchers; see for example refs. [
11,
12,
13,
14]. In this article, instead of considering a generalized derivation, we examine differential identities involving a generalized
-derivation
associated with an
-derivation
∂. Consequently, we prove that either
is a commutative integral domain or
, where
℘ is a prime ideal of an arbitrary ring
ℑ. Furthermore, we explore several sequels and special cases as corollaries of our results. Finally, we devote several examples to emphasize the necessity of the various hypotheses imposed in our theorems.
3. Main Results
In the context of this paper, the pair stands for a generalized -derivation associated with an -derivation ∂, where the two maps are automorphisms on ℑ, unless we mention otherwise. Moreover, the map , defined by for any , expresses the identity map on ℑ.
In ref. [
15] (Lemma 2.1), Bera et al. showed that
∂ maps
ℑ to
, if a semiprime ring
ℑ admits a generalized (
,
)-derivation
associated with an (
,
)-derivation
∂ such that
for every
. Here, we will verify a similar result without imposing any restrictions on
ℑ, as shown below:
Theorem 1. Assume that ∂ is an , β)-derivation on an arbitrary ring ℑ such that for every , where ℘ is a prime ideal of ℑ. Then is a commutative integral domain or the associated -derivation ∂ maps ℑ to ℘.
Proof. The given hypothesis states
applying the linearity in the previous equation, we obtain
if we set
instead of
in Equation (
2) and use it in Equation (
1), we have
again, we set
instead of
in the above relation to obtain
according to the assumption that
℘ is prime and
is an automorphism on
ℑ, we deduce either
or
. Let
and
. Then, it can be easily verified that both
and
are additive subgroups of
ℑ and their union equals
ℑ. Applying Brauer’s trick, we obtain either
or
. If
, then
, for all
and hence
. On the other hand, if
, then
. In the previous relation, as
are automorphisms on
ℑ, it is possible to set
and
to obtain
Thus,
is a commutative integral domain, according to Lemma 1. □
Remark 1. Lemma 2.1 of ref. [
9]
will be a special case of Theorem 1 by putting . The conclusion of ref. [
14], Proposition 1.3, is that either
or
, when
ℑ admits a generalized derivation that satisfies
, for every
, where
℘ is a prime ideal of
ℑ. Also, in ref. [
11], Theorem 3.1, the same conclusion is obtained, when
ℑ admits a multiplicative left-generalized
-derivation associated with an
-derivation
∂ that satisfies
for each
, where
and
are automorphisms of
ℑ. The following theorem aims to discuss the effect of the identity
for any
on the behavior of the ring
ℑ.
Theorem 2. Assume that is a generalized , β)-derivation on an arbitrary ring ℑ such that for every , where ℘ is a prime ideal of ℑ. Then, is a commutative integral domain or the associated -derivation ∂ maps ℑ to ℘.
Proof. For each
, we have the following assumption:
Using the linearity in the previous equation, we obtain
Replacing
with
in the last expression and using it in Equation (
4) gives
For each
, we put
in the place of
in Equation (
5) to obtain
Multiplying Equation (
5) from the left by
and subtracting it from the previous equation, we obtain
Now, applying a similar argument as that after Equation (
3), we obtain the desired conclusion. □
In ref. [
8], Rehman et al. showed that
L is contained in the center of a prime ring
admitting a generalized
-derivation
associated with an
-derivation
∂, that satisfies
for all
, where
L is a Lie ideal and ∗ is an involution on
ℑ. In the following theorem, we will see what happens when the prior identity involves a prime ideal
℘ of a ring
ℑ that is neither prime nor equipped with ∗.
Theorem 3. Assume that is a generalized , β)-derivation on an arbitrary ring ℑ such that for every , where ℘ is a prime ideal of ℑ. Then, is a commutative integral domain or the associated -derivation ∂ maps ℑ to ℘.
Proof. From the given assumption, we have
if
, then
, for all
. The automorphism property of
implies that
, for all
. We set
and
to obtain
, for all
. Hence, using Lemma 1,
is a commutative integral domain.
From now on, let
. Then, for all
, we have
setting
instead of
in Equation (
8) gives
if we multiply Equation (
8) by
from the right and comparing it with the previous equation, we obtain
but
is an automorphism on
ℑ, so the previous equation can be rewritten as
. If we change
by
in the last relation and apply it, we find
. By repeating the similar arguments and techniques after Equation (
3), we obtain the desired result. □
Remark 2. Corollary 11(1) of ref. [
13]
, will be an immediate consequence of Theorem 3 by putting . As an application of the previous theorem, if ℑ is a prime ring, then we have the following corollary:
Corollary 1. Assume that is a generalized , β)-derivation on a prime ring ℑ such that for every . Then, ℑ is either commutative or the -associated derivation ∂ is zero (in this case, ϑ outputs a left centralizer).
The following theorem is an extension of ref. [
7], Theorem 3.5.
Theorem 4. Assume that is a generalized , β)-derivation on an arbitrary ring ℑ such that , for every , where ℘ is a prime ideal of ℑ. Then, is a commutative integral domain or the associated -derivation ∂ maps ℑ to ℘.
Proof. We start with the given assumption
and set
instead of
to have
We multiply Equation (
9) by
from the right and subtract it from the previous equation to obtain
Putting
instead of
in the previous equation and using it give
for each
, this equation is similar to Equation (
3); so, following similar arguments and techniques with some necessary modifications leads to the desired result. □
Theorem 5. Assume that is a generalized , β)-derivation on an arbitrary ring ℑ such that for every , where ℘ is a prime ideal of ℑ. Then, is a commutative integral domain or otherwise the associated -derivation ∂ maps ℑ to ℘.
Proof. The given assumption states that
Linearizing the previous equation and then applying it give for all , that is, for all which is the same as the identity in Theorem 4. Therefore, following it induces the desired conclusion. □
Remark 3. It is easy to verify that, if ϑ is a generalized (α, β)-derivation associated with an (α, β)-derivation ∂, then is also a generalized (α, β)-derivation associated with an (α, β)-derivation ∂.
Applying the previous remark in Theorem 4 leads to the following result:
Theorem 6. Assume that is a generalized , β)-derivation on an arbitrary ring ℑ such that for every , where ℘ is a prime ideal of ℑ. Then, is a commutative integral domain or ∂ maps ℑ to ℘.
Proof. Given that
is a generalized (
,
)-derivation with an (
,
)-derivation
∂, hence, according to Remark 3,
is also a generalized (
,
)-derivation that satisfies Identity
9. Thus,
for each
. Therefore, by employing similar arguments as those mentioned above, we can achieve the desired outcome. □
The question which arises here is whether Theorem 6 is still valid in the case of a commutator. The following theorem provides the answer:
Theorem 7. Assume that is a generalized , β)-derivation on an arbitrary ring ℑ such that for every , where ℘ is a prime ideal of ℑ. Then, is a commutative integral domain or ∂ maps ℑ to ℘.
Proof. Applying arguments and techniques similar to those used to prove Theorem 3 with a few necessary modifications yields the required proof. □
As an application of the previous theorem, we present the following corollary, which is a generalization of ref. [
13], Theorem 6:
Corollary 2. Assume that is a generalized , β)-derivation on an arbitrary ring ℑ such that for every , where ℘ is a prime ideal of ℑ. Then, is a commutative integral domain or ∂ maps ℑ to ℘.
Proof. Note that Remark 3 states that is a generalized (, )-derivation associated with an (, )-derivation ∂. Hence, we can immediately derive the proof by applying the identity for every in Theorem 7. □
Remark 4. In the previous corollary, if we choose both α and β to be equal to the identity map, then ref. [
13]
, Corollary 11(2), is directly taken as a special case. In ref. [
15], Theorem 3.1, Bera et al. discuss the identities
,
and
for all
, where
ℵ is a left ideal of a semiprime ring
ℑ and
are two generalized (
,
)-derivations associated with (
,
)-derivations
, respectively. In the following theorem, without imposing any restrictions on the ring
ℑ, we will discuss analog identities in a prime ideal for one generalized (
,
)-derivation
associated with an (
,
)-derivation
∂.
Theorem 8. Assume that is a generalized , β)-derivation on an arbitrary ring ℑ such that for every , where ℘ is a prime ideal of ℑ. Then, the associated -derivation ∂ maps ℑ to ℘ and
Proof. We have
For each
, we put
instead of
in Equation (
10) to obtain
Multiplying Equation (
10) from the right by
and then comparing with Equation (
11) yields
Once again, if we put
instead of
in Equation (
12), we obtain
Multiplying Equation (
12) from the left by
and then comparing with Equation (
13) yields
that is
Applying the hypothesis that ℘ is prime together with Brauer’s trick, we obtain either for all , or for all .
We begin by assuming that for each
,
. Hence, Equation (
10) is reduced to
for every
. Thus,
for every
. Since
℘ is a prime ideal and
is an automorphism, then
for every
. Therefore,
.
On the other hand, if
,
, then by Theorem 1, either
is a commutative integral domain or
∂ maps
ℑ to
℘. The second case was discussed above, so we consider the case that
is a commutative integral domain. Hence, Equation (
12):
can be rewritten as
. Using the two assumptions that
is an automorphism and
is a prime ideal gives
. Now, we put
instead of
in the last equation and use it to have
for all
. Since
is an automorphism, the previous equation becomes
for all
. Thus, either
or
. Both cases yield
. Therefore, as above, we conclude that
as required. □
In the case that either or ℑ is prime, we derive the following two corollaries, respectively:
Corollary 3. Assume that is a generalized derivation on an arbitrary ring ℑ such that for every , where ℘ is a prime ideal of ℑ. Then, ∂ maps ℑ to ℘ and .
Corollary 4. Assume that is a generalized , β)-derivation on a prime ring ℑ such that for every . Then, the associated -derivation ∂ is zero and
Theorem 9. Assume that is a generalized , β)-derivation on an arbitrary ring ℑ such that for every , where ℘ is a prime ideal of ℑ. Then, is a commutative integral domain.
Proof. According to the given hypothesis, we have
Setting
instead of
in Equation (
14) gives
Multiplying Equation (
14) from the right by
and then comparing with Equation (
15) yields
If we set
instead of
in Equation (
16), we obtain
Now, multiplying Equation (
16) from the left by
and comparing with Equation (
17) yields
The automorphism property of
gives
where
℘ is prime, and applying Brauer’s trick implies either
for all
, or
for all
. We start with the first case when
for all
and apply Theorem 1, which implies that either
is a commutative integral domain or
∂ maps
ℑ to
℘. When
∂ maps
ℑ to
℘, then Equation (
14) can be reduced to
for every
. We set
instead of
in the previous equation and apply some calculations to obtain
for all
, which is equivalent to
for any
. Since
is an automorphism on
ℑ, the previous expression becomes
for any
, which means that
. As
℘ is prime and does not equal to
ℑ, then the last relation becomes
for any
. Then, applying Lemma 1 implies that
is a commutative integral domain which completes the proof. □
Corollary 5. Assume that is a generalized derivation on an arbitrary ring ℑ such that for every , where ℘ is a prime ideal of ℑ. Then, is a commutative integral domain.
Corollary 6. Assume that is a generalized derivation on a prime ring ℑ such that for every . Then, ℑ is commutative.
Theorem 10. Assume that is a generalized , β)-derivation on an arbitrary ring ℑ such that for every , where ℘ is a prime ideal of ℑ. Then, is a commutative integral domain.
Proof. By this hypothesis, we have
for any
, set
instead of
in Equation (
18) to have
Then, multiplying Equation (
18) from the right by
and then comparing with Equation (
19) yields
Choose
in Equation (
20) to obtain
The previous equation is similar to Equation (
16), so we repeat similar arguments and techniques to obtain the desired goal. □
We can derive the following two corollaries:
Corollary 7. Assume that is a generalized derivation on an arbitrary ring ℑ such that for every , where ℘ is a prime ideal of ℑ. Then, is a commutative integral domain.
Corollary 8. Assume that is a generalized derivation on a prime ring ℑ such that for every . Then, ℑ is a commutative.
Finally, we devoted the following examples to emphasize the necessity of the various hypotheses imposed in our theorems:
Example 1. Let ℑ be the quaternions ring , that isand let be a prime ideal of the quaternions ring . Define bywhere . It is easy to see that is a generalized -derivation associated with ∂ where α and β are automorphisms on ℑ. Furthermore, we can see that and for every , although is not commutative and . Example 2. Let Λ be a ring with property and let . Since and , then and for any . Let . Define by , and . It is easy to check that is a generalized -derivation of ℑ. Furthermore, , , , , , , for every . Also, we note that is not an integral domain and ℘ is not a prime ideal as , but neither nor . Also, α and β are not automorphisms whenever .
Example 3. Let be the ring of polynomial with integers coefficient, and let . Define by , , and . It is easy to verify that is a generalized -derivation associated with ∂. One can check that , , , for every , although neither is an integral domain nor . Also, ℘ is not a prime ideal in , since , but as well as α and β are not automorphisms. So, the primeness of ℘ and the automorphism property of α and β are necessary conditions.
Example 4. Let , where is the Hamilton ring as in Example 1 and let . Define by , , , and . It is easy to verify that is a generalized -derivation associated with an -derivation ∂. Also, one can check that , , , , , , , , , for all . However, neither commutative nor ∂ maps the ring ℑ to a prime ideal ℘. Note that ℘ is not a prime ideal of ℑ since but .