On Commutativity of Rings Under Certain Polynomial Constraints

The pioneer theorem of Weddernburn on commutativity of division rings was proved in the very beginning of twentieth century. Aside from its own intrinsic beauty and important role in many diverse parts of algebra specially, the theorem serves as the starting point for investigations of certain kinds of conditions that render a ring commutative. A large part of the results in this area was developed in the hands of many distinguished mathematicians like Jacobson, Herstein, Kaplansky, Faith, Martindale, Nakayama, Bell and many others. The purpose of the present paper is to investigate commutativity of a ring with unity 1 satisfying certain polynomial constraints. The main result of the first section asserts that a ring is commutative if at least one of the integral exponent used in the polynomial constraints of the theorem is zero and the ring also satisfies the property Q(n) Further, in the second section, commutativity of a ring with unity 1 has also been established under a set of different polynomial identities applying the most frequently used technique known as Streb’s classification. Finally, in the last section, these results of the foregoing sections are further extended to a special class of rings called as one sided s unital rings.


Introduction
Throughout the present paper, R will denote an associative ring (may be without unity 1), ( ) C R the commutator ideal, ( ) P R the center and ( ) N R the set of nilpotent elements of a ring R respectively. As usual, [ ] Z X is the totality of polynomials in X with coefficients in , Z the ring of integers. For any , x y R ∈ , the symbol [ , ] x y stands for commutator . xy yx − A ring R is said to have the property ( ) Q n if for any positive integer n and for all , x y R ∈ , [ , ] 0 n x y = ⇒ [ , ] 0 x y = . The present paper contains 4 sections. Section 1 deals with introduction, section 2 includes commutativity of ring with unity satisfying some identities, section 3 incorporates commutativity of ring with unity through Streb's classification and satisfying some other related identities and finally the results of the foregoing sections are extended for s -unital rings in section 4. The famous theorem due to Jacobson [17] asserts that if every element x of a ring R satisfies the condition ( ) , n x x x = where ( ) 1 n x > is a positive integer, then R is commutative.
This result at the same time generalizes that every finite division ring is commutative and also the result that every Boolean ring is commutative. Among the interesting generalizations of Jacobson's theorem, Herstein [13] proved that rings satisfying the polynomial identity ( ) n n n x y x y + = + for some 1 n > must have nil commutator ideal. Among other classes of rings in which ideal is known to be nil is the class of rings satisfying the polynomial identity [ , ] [ , ] n n x y x y = for some 1 n > . It can be easily observe that this class includes the rings satisfying the identity . ( ) .
n n n x y x y + = + Bell [8] proved the commutativity of a ring R with unity 1 satisfying the polynomial identity [ , ] [ , ] n n x y x y = if the additive group ( , ) R + is n-torsion free. Motivated by the above observations, the main purpose of this paper is to investigate A number of authors have studied commutativity of rings satisfying various special cases of the above property. The objective of the present section is to generalize these results for rings with unity 1 satisfying the above property. Further, the results are extended for one sided s-unital rings in the subsequent section.

Commutativity of Ring with Unity
Main result of this section states as follows: x y y x y x y x + + = This is a polynomial identity and it can be easily observe that 11 21, 11 x e e y e = − + = fail to satisfy this equality in 2 ( ( )) , GF p where p is a prime. Hence, by Lemma 2.
. On the other hand if 0, s = then choose 11 12, 11 x e e y e = − + = to get the required result.
Thus, application of (2) gives  x y The following corollary is an immediate consequence of the above theorem.
fixed non-negative integers and let R be a ring with unity 1 in which for every y R ∈ there exist integers ( ) 0, for all x R ∈ . Further, if at least one of , r s is zero and R satisfies the property ( ) Q n , then R is commutative.
The existence of unity 1 in the hypothesis of Theorem 2.1 can be justified by the following example.
Example 2.1 Let k D be the ring of all k k × matrices over a division ring D and

Then 3 A is a non-commutative ring of index 3 which satisfies the identity [ , ] [ , ]
n n x y x y = for all 3 , x y A ∈ and 2. n ≥ The following example strengthens the existence of the property ( ) Q n in the hypothesis of the Theorem 2.1.
Then R is a non-commutative ring with unity 1 satisfying the identity [ , ] [ , ] n n x y x y = for all , x y R ∈ and 2. n =

Commutativity of Rings Through Streb's Classification
In an attempt to generalize a well known theorem due to Bell [7], Quadri and Khan [31] proved; a ring R with unity 1 is commutative if it satisfies a polynomial identity  [6] established that the above result remains true if the value of the exponent m appearing in the given identity is no longer fixed rather depends on the ring element y . Recently, Nishinaka [27] improved this result as follows: a ring R with unity 1 is commutative if it satisfies the condition [ , where m and n are fixed non-negative integers. The objective of the present paper is to further extend the study in this direction and investigate the commutativity of a ring R satisfying the following properties: x y R there exists an integer 0 m ≥ and The discussion starts with the following theorem: Theorem 3.1 Let R be a ring with unity 1 satisfying either of the property 1 ( ) P or 2 ( ) P , then R is commutative (and conversely).
The following rings should be taken into consideration in order to develop the proof of the above theorem: a non-commutative radical subring of S such that [ , ] [ , ] 0.

T T T T T T = =
In the year 1989, Streb [33] had given the classification of non-commutative rings which has been used effectively as a tool to obtain a number of commutativity theorems (cf. [22], [23], [24] further references can be found). From the proof of [34, corollary 1], it can be easily observe that if R is a non-commutative ring with unity 1, then there exists a factor subring of R which is of the form ( ), ii ( ), iii ( ) iv or ( ) v . This observation gives the following result which plays the key role in our subsequent study (cf., [24,Lemma 1]. Lemma 3.1 Let P be a ring property which is inherited by factor subrings. If there no rings of the form ( ) i , ( ), ii ( ), iii ( ) iv or ( ) v satisfy the property P , then every ring with unity 1 satisfying the property P is commutative.
The following result has been proved in [22,Corollary 1]. Lemma 3.2 Suppose that a ring R with unity 1 satisfies the condition ( ) CH and if R is a non-commutative ring, then it is always possible to find out a factor subring of R which is of the form ( ) i or ( ). ii The following result is due to Herstein [13]. Lemma 3.3 Let R be a ring in which for every , The proof of the following lemma is essential for developing the proof of the main theorem.
Lemma 3.4 If R is a division ring satisfying either of the property 1 ( ) P or 2 ( ) P , then R is commutative. Proof. Suppose R satisfies the property 1 ( ) P . Let u be a unit in R , then for every y R ∈ there exist polynomials as a finite field with a non-trivial automorphism. Now, choosing   If R is a ring of the form ( ) ( )

G F Then 2
R is a noncommutative right s -unital ring satisfying the property [ , ] [ , ] r n t m j x x y y x y y = for any fixed positive integers R has the property ( ) Q n for odd . n Before moving ahead to establish commutativity of sunital rings satisfying some related properties as considered in Theorem 3.1, the following lemma is due to Komatsu et al. [25] is pertinent in order to make our paper self-contained. Lemma [15], it can be assumed that R has unity 1 and commutativity of R follows by Theorem 3.1.
Using the similar arguments as used to get Theorem 3.2, the following can be easily proved: n n x y x y = if the additive group ( , ) R + is n-torsion free. This result of Bell possess a natural question that weather it is feasible to extend this result for some wider polynomial identities or not. This paper gives an affirmative answer of this question and hence establishes commutativity of a ring with unity 1 satisfying some wider polynomial identities and further assures the commutativity of one sided s − unital rings also. The same results may also be extended to another class of rings called near rings as well as derivations of rings. s s s