Commutativity of Prime Ring with Orthogonal Symmetric Biderivations

Published online: March 6, 2019 The intention of the present research article is to generalize the performance of prime rings (commutativity) with certain algebraic identities using jordan ideals. Familiar results characterizing commutativity of prime ring with orthogonal biderivations have been discussed here with Jordan ideals. Whenever some biderivations of prime ring satisfying certain commutator relations B u B v 1 2 , , , , v w u w ( ) ( )   = [ ] , B v B w B w B v 3 1 2 3 , , , , , w u u w u v ( ) ( )( ) ( )= [ ] , for all u, , v w J ∈ then that ring is commutative.


Introduction
More than a few authors, investigated the structure prime ring & semiprime rings (commutativity) accepting the derivations, generalized derivations etc. The notion of derivations of prime rings was originated by (Posner 1957), jordan derivations of prime rings was originated by (cusack 1975). These derivations was extended by (Bell and Daif 1995) for commutativity of prime rings. Later on (Bresar 1993) used centralizing concept using derivations. These generalizations was done in the article derivations using semiprime rings with results are commutative by (Daif 1998). The concept of symmetric biderivations on prime and semiprime rings was introduced by (Vukman 1989).The notation and terminology in this paper follows (Vukman 1990 andOukhtite 2011). Many authors have their contribution to orthogonality of derivations on semiprime as well as prime rings. The idea of orthogonality of derivations on semiprime as well as prime rings was developed by (Vukman and Bresar 1989). (Argac 2004) studied orthogonality conditions for generalized derivations. (Ashraf 2010) obtained the orthogonality conditions for a pair of derivations in gamma rings. with their results (Jaya Subba Reddy et. al. 2016) obtained the essential and sufficient conditions of biderivations to be orthogonal. (Oukhtite et. al. 2014) proved the commtativity results of prime rings with derivations using jordan ideals. In this current study it was extended the results of commutativity of prime rings with orthogonal biderivations using Jordan ideals. In the present article we studied some theorems related to commutativity of prime rings using commutator identities satisfied by biderivations with Jordan ideals. We established the following theorems as follows.

Preliminaries
In each part of this article all rings assumed to be associative and possesses an identity. As a well-known the commutator (uvvu) will be symbolized as [u, v]. We are wellknown that R is a prime ring if uRv = 0 ⇒ u = 0 or v = 0 and is semiprime if uRu = 0 ⇒ u = 0. If D (uv) = D(u) v + vD(u), for any u, v R ∈ then we call this additive map D R R : → is a derivation. We Defined, biadditive mapping B R R R .,. :  Vukman and Bresar 1989). Likewise, any pair (B, D) of biderivations are said to be orthogonal if B u v RD v r Dv r RB u v , , , , , then we say J is a Jordan ideal of R. Note that B(x) means B(x,m) means for some m J ∈ . In the entire paper R act as a prime ring with 2-torson free & J ≠ 0 is a jordan ideal of R Following are known results to the readers Res 1: If a u , 2 0     = for any u J ∈ , then is in center of R.

Res 2:
If an additive subgroup is a subset of Z(R), then R is commutative ring. Res 3: a non commutative ring R satisfies a u vw b , , for every v w J , ∈ , u R ∈ , then a = 0 or b = 0. We studied the following lemmas for proving the main theorems Lemma 2.1 (Reddy C.J.S and Reddy B.R 2016) A semiprime ring R of characteristic not two, a pair of biderivations B 1 and B 2 are to be orthogonal ⇔ the following results are equivalent: , ∈ , then orthogonality of B 1 and B 2 are satisfied, also either B 1 = 0 or B 2 = 0.
By using lemma 2.1 B 1 , B 2 , are orthogonal, that is Put [s, pq]y instead of v, for any p,q ∈ ∈ J s R , in the equation (2) and use (2), to get Replace v by vt for some t J ∈ in the equation (3), to obtain (4) Assume that R is commutative, then replacing u by u 2 in the equation (1) (7) If R is commutative, substitute u by u 2 in equation (7), we get Already a well known result, from the definition, a prime ring itself an integral domain, so the equation (8)  (9) From lemma 2.1, B 1 and B 2 are orthogonal.
One can see that the equation (9) is same as compared with equation (2) so using the above lemma, we conclude B u

Lemma 2.4
Any two biderivations B 1 and B 2 satisfies the condition , , ∈ , then either B 1 0 = or B 2 0 = , and also R is commutative.

Proof: Consider for every
Replacing w by wm, for any m J ∈ , in the equation (10) From equation (12) and equation (13), we get B u wB v So B u 1 , v ( ) is commuting, then Bresar (Bresar 1993), gives that R is commutative and equation (10)

Main Theorems
By replacing w with wm, m ∈ J in the Eq. (16), to get (17) By replacing w with w [s, pq] in the equation (17), to obtain   (22) and equation (17) are identical, continuing the procedure as we done in the theorem 3.1, it is clear that R is commutative.
Replacing u with tu 2 in the equation (25),to get Replacing t with B 3 (v, w)t in equation (26) It is clear that (27) and (20) are identical, hence we conclude that R is commutative then equation (23) becomes