Ideals and Symmetrc Left Bi-Derivations on Prime Rings

 

Dr. C. Jaya Subba Reddy, G. Venkata Bhaskara Rao

Department of Mathematics, Sri Venkateswara University, Tirupati-517502, Andhra Pradesh, India.

 

ABSTRACT:

Let  be a non commutative 2, 3-torsion free prime ring and  be a non zero ideal of . Let  be a symmetric left bi-derivation such that  and  is a trace of . If (i), for all , (ii) , for all , then . Suppose that there exists symmetric left bi-derivations  and  and  is a symmetric bi-additive mapping, such that (i) , for all , (ii) , for all , where   and   are the traces  of  and  respectively and  is trace of , then either  or . If  acts as a left (resp. right) -homomorphism on , then .

 

KEYWORDS:

 

 

INTRODUCTION:

The concept of a symmetric bi-derivation has been introduced by Maksa.Gy in [5],[6]. A classical result in the theory of centralizing mappings is a theorem first proved by Posner.E [7]. Vukman.J [8] has studied some results concerning symmetric bi-derivations on prime and semi prime rings. Yenigul.M.S and Argac.N [9] proved that the results which are obtained in [8, Theorems 1, 2, 3] by using a nonzero ideal of . Jaya SubbaReddy.C [2], [3] has proved some results concerning symmetric left bi-derivation on prime rings. In this paper we proved some results concerning to ideals and symmetric left bi-derivations on prime rings.

 

Throughout this paper  will be associative. We shall denote by  the center of a ring . Recall that a ring  is prime if  implies that or . We shall write for  and use the identities, . An additive map  is called derivation if , for all.

 

A mapping  is said to be symmetric if , for all. A mapping  defined by , where  is a symmetric mapping, is called a trace of B. It is obvious that, in case  is symmetric mapping which is also bi-additive (i. e. additive in both arguments) the trace of  satisfies the relation , for all.We shall use also the fact that the trace of a symmetric bi-additive mapping is an even function. A symmetric bi-additive mapping  is called a symmetric bi-derivation if , for all . Obviously, in this case also the relation , for all. A symmetric bi-additive mapping  is called a symmetric left bi-derivation if , for all . Obviously, in this case also the relation , for all . A mapping is said to be commuting on if , for all . A mapping  is said to be centralizing on if , for all . A ring  is said to be n-torsion free if whenever , with , then , where  is nonzero integer. Let  be a ring and  be a non zero left (right) ideal of .

We shall say that the mapping  acts as a left (right) -homomorphism on  if  and  (resp. and ) for all  and . Let  be a set  (resp. )  will denotes the left (resp. right) annihilator of .

 

Lemma 1:[4, Lemmas 2.1 and 2.2] Let  be a derivation on a prime ring  and  a non zero ideal of . Suppose that either (i) , for all  or (ii) , for all  holds. Then or .

Proof:

(i). We have , for all .                                                                                                                                (1)

We replace  by  in (1), we get

, for all . 

By using (1) in the above equation, we get

, for all.

Since  is prime which implies that either or .

(ii) We have , for all .                                                                                                                                (2)

We replace  by  in (2), we get

, for all .  

By using (2) in the above equation we get

, for all .  

Since  is prime which implies that either or .

Lemma 2:[1, Lemma 1] Let  be a prime ring and  a non zero ideal of . If  is commutative then  is commutative.

Lemma 3: Let  be a 2-torsion free prime ring and  a non zero ideal of . Let  be fixed elements of . If , for all , then either  or .

Proof: We have , for all .       

 , for all .                                                                                                                                    (3)

We replace  by  in (3), we get

By using (3) in the above equation, we get

Again by using (3) in the above equation, we get

, for all .

Since  is a 2-torsion free prime ring, we get

, for all  and hence or .

Lemma 4: Let  be a 2-torsion free prime ring and  a non zero right (or left) ideal of . Let  be a symmetric left bi-derivation and  be a trace of . Suppose that , for all , then , that is .

Proof: We have , for all .                                                                                                              (4)

We replace  by  in (4), we get

By using (4) in the above equation we get

Since  is a 2-torsion free ring, we get

, for all .                                                                                                                                  (5)

We replace  by  in (5), where

By using (5) in the above equation, we get

, for all and .

Since the right annihilator of non zero right ideal is zero, we have

, for all and .                                                                                                                      (6)

We replace  by  in (6), we get

By using (6) in the above equation, we get

, for all .

Since the right annihilator of non zero right ideal is zero, we have

, for all .

 

Theorem 1: Let  be a non commutative 2-torsion free prime ring and  be a non zero ideal of . Let  be a symmetric left bi-derivation such that  and  is a trace of . If , for all , then .

Proof: Since  is non commutative, this implies  is non commutative ideal of  by lemma 2. Since  is non zero ideal of 2-torsion free prime ring ,  itself is a non commutative 2-torsion free prime ring. Therefore, for all  by [2, Theorem 1] and , for all  by lemma 4.

 

Theorem 2: Let  be a non commutative 2-torsion free and 3-torsion free prime ring and  be a non zero ideal of . Let  be a symmetric left bi-derivation such that  and  is a trace of . If , for all ,  then .

Proof: Since  is non commutative, this implies  is non commutative ideal of  by lemma 2. Since  is non zero ideal of 2-torsion and 3-torsion free prime ring ,  itself is a non commutative 2-torsion and 3-torsion free prime ring. We have , for all ,  by the proof of [2, Theorem 2]. Hence , for all  by Theorem 1.

 

Theorem 3: Let  be a 2-torsion free prime ring and  be a nonzero ideal of R.  Suppose that there exists symmetric left bi-derivations and  such that , for all , where  denotes the trace of . Then either or .

Proof: It is enough to show that  or    by lemma 4. We have in the proof of [3, Theorem 1]

, for all .                                                                                       (7)

And , for all .                                                                                   (8)

Suppose that and , then there exist  such that   and .

In particular , for all  by (8).

Since , we have , by lemma 3. Similarly we get .

We replace  by  in (7), we get

, for all .

By using lemma 1 in the above equation we get

, for all , since .

In particular  in the above equation we get .

In the same way, we can get .

Let us write  for

then . Similarly we can obtains . But  and  cannot be both different from zero according to (8) and lemma 3. Therefore we have proved that either or . Which is actually assertion of the theorem.

 

Theorem 4: Let  be a 2-torsion and 3-torsion free prime ring and  be a nonzero ideal of R.  Suppose that there exist symmetric left bi-derivations and . Suppose further that there exists a symmetric bi-additive mapping  such that , for all , where   and   are the traces  of  and  respectively and  is trace of . Then either or .

Proof: It is enough to show that , for all . We have in the proof of [3, Theorem 2] , for all .

Let us assume that , for some  in the above equation, then  according to lemma 1, contrary to assumption , therefore , for all . The proof of the theorem is complete since all the requirements of the theorem 3 are fulfilled.

 

Theorem 5: Let  be a ring and  be a non zero left (resp. right) ideal of  such that  (resp. ). Let  be a symmetric left bi-derivation. If  acts as a left (resp. right) -homomorphism on , then .

Proof: Suppose that  is a left ideal such that  and  acts as a left -homomorphism on . Then .

, for all and .

Hence , for all  and .

This implies , for all  and .                                                                                              (9)

We replace  by  in (9), where , we get

By using (9) in the above equation we get , for all  and .

Hence  for all .

This implies  for all , hence .

Theorem 6: Let  be a prime ring and  be a non zero left (resp. right) ideal of . Let  be a symmetric left bi-derivation. If  acts as a left (resp. right) -homomorphism on , then .

Proof: Suppose that  is a left ideal of  and   acts as a left -homomorphism on . Then

, for all and .                                                                                                                              (10)

We replace  by  in (10), we get

, for all and .                                                                                          (11)

By using (10) in (11), we get

This implies that , for all and .

By lemma 1, we get , for all .

By lemma 4, we get . 

 

REFERENCES:

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4.       Kaya K. Prime rings with -derivations, Hacettepe Bulleten of Noth. Sci. and Eng. (1987); 16: 63-71.

5.       MaksaGy. A remark on symmetric bi additive functions having nonnegative diagonalization. Glasnik Mathematics. (1980); 15(35): 279 – 282.

6.       MaksaGy. on the trace of symmetric bi-derivations. C. R. Math. Rep. Acad. Sci. Canada (1987); 9: 303 – 307.

7.       Posner E: Derivations in prime rings. Proceedings of American Mathematical Society.   (1957); 8:1093- 1100.

8.       Vukman J. Symmetric bi-derivations on prime and semi prime rings. Aequationes mathematics. (1989); 38: 245 – 254.

9.       Yenigul MS and Argac N. Ideals and symmetric bi-derivations of prime and semiprime rings. Mathematical Journal of Okayama University.(1993); 35:189-192.

 

 

 

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