An identity involving automorphisms of prime rings inspired by Posner's theorem

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.

KEYWORDS:

• Prime ring
• Lie ideal
• automorphism
• polynomial identity
• centralizing mapping

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16N60
• 16W20

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 [1], 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 [2] proved that a simple artinian ring is commutative if it has a commuting automorphism different from the identity mapping. In [3], 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 [4], where further references can be found). Later, Mayne [5] 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 [6] 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 [7] 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 [8] 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:

1. There exists such that for all .

2. 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 [9, 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 ([10, 11]):

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 ([12, Lemma 2.1]):

Let be a prime ring with extended centroid . Then the following conditions are equivalent:

1. .

2. satisfies , the standard in four variables.

3. is commutative or embeds in , where is a field.

4. is algebraic of bounded degree 2 over .

5. 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 (1(1) ) is a non-trivial generalized polynomial identity on . Hence by [13, Theorem 2], is also an identity for . By Martindale's theorem [14], 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 [15], satisfies . Since -word degree is 2 and we have either or , hence by [15, Theorem 3] we infer that satisfies the following identityNote 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 [16, 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 [9, 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 havefor 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 thatSo that we haveand hencewhich 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 thatSuppose σ is -inner automorphism. Then there exists an invertible element such that for all . Thus satisfiesBy a theorem of Chaung [13], and satisfy the same generalized polynomial identities. Thus satisfiesSince , 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 [17, Theorem 3.5]. Clearly . So we may regard as a subring of . Let , the maximal right ring of quotients of . By [1, Theorem 6.4.4], is also a non-trivial generalized polynomial identiy on . By Martindale [14], , 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 [13, 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 [15, 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 [16, 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