ABSTRACT
Let be a prime ring with centre
,
a non-zero Lie ideal of
, and σ a non-trivial automorphism of
such that
for all
. If
, then it is shown that
satisfies
, the standard identity in four variables.
1. Introduction
Throughout, will represent an associative ring with the centre
,
will be the maximal right ring of quotients of
and
, the Utumi quotient ring of
. The centre of
, denoted by
is called the extended centroid of
(we refer the reader to [Citation1], for the definitions and related properties of these objects). For any
, the symbol
stands for the commutator xy−yx. Note that a ring
is prime if
implies either x=0 or y=0. An additive mapping
is called a derivation on R if
holds for all
. Further, an additive mapping
is called a generalized derivation associated with a derivation
if
holds for all
. For a non-empty subset S of
, a mapping
is said to be centralizing on S, if
for all
. In particular if
for all
, then
is said to be commuting on S. The study of centralizing and commuting mappings goes back to 1955, when Divinsky [Citation2] proved that a simple artinian ring is commutative if it has a commuting automorphism different from the identity mapping. In [Citation3], E.C.Posner studied the centralizing derivations of prime rings and proved that a prime ring
must be commutative if it admits a non-zero centralizing derivation on
. This theorem due to Posner has turned to be very influential in the field of Ring Theory. A number of authors have refined and extended this theorem in several ways (see [Citation4], where further references can be found). Later, Mayne [Citation5] obtained the analogous result for automorphisms. To be more precise, he proved that if a prime ring
admits a non-trivial automorphism or a derivation, which is centralizing on a non-zero ideal of
, then it must be commutative. This result was further extended by Mayne [Citation6] itself and obtained the similar conclusion for a prime ring admitting a non-trivial automorphism which is centralizing on a non-zero quadratic Jordan ideal of
. Recently, Cheng [Citation7] studied derivations of prime rings that satisfy the special type engel condition. More precisely, he proved that if
is a prime ring of characteristic different from 2 which admits a non-zero derivation δ such that
, for all
, then
must be commutative.
More recently, De Filippis and Tammam El-Sayiad [Citation8] obtained the generalization of the above result. To be more specific, they proved the following result:
Theorem 1.1:
Let be a prime ring with
a non-zero generalized derivation of
a non-central Lie ideal of
. If
for all
then one of the following holds:
There exists
such that
for all
.
satisfies the standard identity
and there exists
such that
for all
.
Motivated by this result, we obtained the analogous theorem for automorphisms of prime rings. In fact, it is shown that if is a prime ring of characteristic different from two which admits a non-trivial automorphism σ such that
for all
, a non-central Lie ideal of
, then
satisfies
, the standard identity in four variables.
2. Preliminaries
In this section, we first fix some notations and definitions. The standard identity in four variables is defined as follows:
where
is a sign of permutation μ of the symmetric group of degree 4. Recall that throughout this paper
will always denote a prime ring and
is the maximal right ring of quotients of
. We also know that any automorphism of
can be uniquely extended to an automorphism of
. An automorphism σ of
is called
-inner if there exists
such that
for every
. Otherwise σ is called
-outer. we denote by
, the group of all automorphisms of
and by
, the group consisting of all
-inner automorphisms of
. Note that a subset
of
is independent (modulo
) if for any
,
implies
. For instance, if a is an outer automorphism of
, then 1 and a are independent (modulo
.
We also denote by , the right vector space over a division ring
and
will denote the ring of
-linear transformations on
. A map
is called a semi-linear transformation if
is additive and there is an automorphism ϕ of
such that
for all
and
. Moreover, by the theorem of Jacobson [Citation9, Isomorphism Theorem, p.79], there exists an invertible semi-linear transformation
such that
for all
, where σ is an automorphism of
. We mention some important known results which will be useful for developing the proof of our main result.
Fact 2.1 ([Citation10, Citation11]):
Let be a prime ring and
a non-central Lie ideal of
. If
, then there exists a non-zero ideal
of
such that
. If
and
, then there exists a non-central ideal
of
such that
. Thus if either
or
, then we may conclude that there exists non-zero ideal
of
such that
.
Fact 2.2 ([Citation12, Lemma 2.1]):
Let be a prime ring with extended centroid
. Then the following conditions are equivalent:
.
satisfies
, the standard in four variables.
is commutative or
embeds in
, where
is a field.
is algebraic of bounded degree 2 over
.
satisfies
.
3. The results in prime rings
We facilitate our discussion with the following lemmas which will be used to obtain the proof of our main result.
Lemma 3.1:
Let be a prime ring with characteristic different from 2 and σ be a non-identity automorphism of
such that
for all u in a non-central Lie ideal of
then
satisfies
the standard identity in four variables.
Proof:
We assume that . In the light of the Fact 2.1, there exits a non-zero ideal
of
such that
. So by our hypothesis, we have
(1) Firstly, if σ is
-inner, then there exists an invertible element
such that
for all
. Since
, so relation (Equation1
(1) ) is a non-trivial generalized polynomial identity on
. Hence by [Citation13, Theorem 2],
is also an identity for
. By Martindale's theorem [Citation14],
is a primitive ring with non-zero socle. Since
is a primitive ring, there exists a vector space
over a division ring
such that
is a dense ring of
- linear transformations over
. To this end, we divide the proof into two cases:
Case 1. Here our intention is to show that for any , v and bv are linearly
-dependent, so we may assume that
. If v and bv are linearly
-independent for some
, then we consider the following possibilities:
If , then the set
is linearly
-independent. By the density of
, there exists
, such that
so that
We observe that
On the other hand if
, then
for some
. Again invoking the density theorem we find that, there exist
, such that
Again we see that
for some
, a contradiction.
Case 2. We have that v and bv are -dependent for every
. For every
, we write
where
. Fix
. Suppose that v and u are
-independent. Then
. So
, and hence
. Suppose that u and v are
-dependent. As
, so for any
, w and u are
-independent and then by same argument as above, we have
. Thus
is independent of
. That is, there exists
such that
for all
. Hence for any
,
and
. Therefore we have
for any
. i.e.
, a contradiction. Therefore
implying that
is commutative which contradicts our hypothesis
. This forces us to conclude that
and hence by using the Fact 2.2,
satisfies
.
Next, we assume that σ is not -inner. Then by the Chuang [Citation15],
satisfies
. Since
-word degree is 2 and we have either
or
, hence by [Citation15, Theorem 3] we infer that
satisfies the following identity
Note that this is a polynomial identity and thus there exists
such that
, the ring of
matrices over
, where
. Moreover,
and
satisfy the same polynomial identity [Citation16, Lemma 1], that is
. Let
be a matrix unit with 1 in the
-entry and zero elsewhere. By choosing
, we get
. This forces us to conclude that K=1, i.e.
is commutative, a contradiction. This completes the proof.▪
Lemma 3.2:
Let σ be an automorphism of such that for every
. If
then σ is an identity map on
.
Proof:
Using a theorem of Jacobson [Citation9, Isomorphism Theorem, p.79], there exists a semi-linear transformation such that
for all
. In particular, there exists an automorphism ϕ of
such that
for all
and
. Now by our hypothesis we have
for all
. Here also, two cases arise. Suppose there exists
such that v and
are
-independent. If
is
-independent, then by density theorem there exists
such that
So that we have
and hence
which is a contradiction.
Next suppose that is
-dependent. Then there exists
such that
. Again let
such that
so that we have
Thus we see that
which is again a contradiction.
Next we have that v and are
-dependent for every
. For each
, we write
, where
. Fix
. Let
and
. Suppose first that u and v are
-independent. Then
. So
and hence
. Suppose that u and v are
-dependent. Since
, there exists
such that w and u are
- independent and then by the proof above, we have
. Clearly w and v are
-independent and hence
implies that
. Thus
is independent of the choice of
. Consequently,
for all
, where
. Thus
for all
and hence
for all
and
. In particular,
for all
. Thus
for all
. This implies that σ is an identity map on
, which proves the lemma.▪
Now, we are ready to prove our main theorem.
Theorem 3.1:
Let be a prime ring,
a non-central Lie ideal of
and σ a non-identity automorphism of
such that
for all
. If
then
satisfies
the standard identity in four variables.
Proof:
Let us suppose that . Then by the Fact 2.1, there exists a non-zero ideal
of
such that
. By our hypothesis, this gives that
Suppose σ is
-inner automorphism. Then there exists an invertible element
such that
for all
. Thus
satisfies
By a theorem of Chaung [Citation13],
and
satisfy the same generalized polynomial identities. Thus
satisfies
Since
,
for all
is a non-trivial generalized polynomial identity on
. Denote by
, the algebraic closure of
. If
is infinite and set
for
finite. Then
is a prime ring with extended centroid
[Citation17, Theorem 3.5]. Clearly
. So we may regard
as a subring of
. Let
, the maximal right ring of quotients of
. By [Citation1, Theorem 6.4.4],
is also a non-trivial generalized polynomial identiy on
. By Martindale [Citation14],
, where
is a vector space over a division ring
and
is of finite dimension over its centre
. Recall that
is either algebraic closed or finite. From the finite dimentionality of
over
, it follows that
. Hence
. By Lemma 3.2, we get a contradiction.
We now assume that σ is a -outer automorphism, by Chuang [Citation13, Main Theorem]
and
satisfy the same polynomial identities and hence
does as well. Therefore
satisfies
. Since
-word degree is 2 and we have either
or
, again by [Citation15, Theorem 3] it follows that
for all
. Note that, this is a polynomial identity and thus there exists a field
such that
, the ring of
matrices over a field
, where
. Moreover,
and
satisfy the same polynomial identities [Citation16, Lemma 1], that is,
for all
. Let
be the matrix unit with 1 in the
-entry and zero elsewhere. Since
, so
. By choosing
, we get
, a contradiction. Thus
and hence by the Fact 2.2, we get the required result, which completes the proof of the theorem.
Finally, we end up with the following example, which shows that primeness condition on is not superfluous in our hypothesis.▪
Example 3.1:
Let denote the ring of all
matrices over a field
satisfies
. Let
and
. Then
is semi-prime ring and
is a non-zero Lie ideal of
. We define
by
. It can be easily seen that σ is an automorphism which satisfies
for all
.
Disclosure statement
No potential conflict of interest was reported by the authors.
ORCID
Sajad Ahmad Pary http://orcid.org/0000-0003-3542-0219
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