Skew-Commuting Derivations of Noncommutative Prime Rings

The main purpose of this paper is study and investigate a skew-commuting and skew-centralizing d and g be a derivations on noncommutative prime ring and semiprime ring R, we obtain the derivation d(R)=0 (resp. g(R)=0 ) .


Introduction
Derivations on rings help us to understand rings better and also derivations on rings can tell us about the structure of the rings. For instance a ring is commutative if and only if the only inner derivation on the ring is zero. Also derivations can be helpful for relating a ring with the set of matrices with entries in the ring (see, [5]). Derivations play a significant role in determining whether a ring is commutative, see ( [1], [3], [4], [18], [19] and [20]).Derivations can also be useful in other fields. For example, derivations play a role in the calculation of the eigenvalues of matrices (see, [2]) which is important in mathematics and other sciences, business and engineering. Derivations also are used in quantum physics(see, [18]). Derivations can be added and subtracted and we still get a derivation, but when we compose a derivation with itself we do not necessarily get a derivation. The history of commuting and centralizing mappings goes back to (1955) when Divinsky [6] proved that a simple Artinian ring is commutative if it has a commuting nontrivial automorphism. Two years later, Posner [7]has proved that the existence of a non-zero centralizing derivation on prime ring forces the ring to be commutative (Posner's second theorem).Luch [8]generalized the Divinsky result, we have just mentioned above, to arbitrary prime ring. In [9] M.N. Daif [12] , give some results as , let R be a 2-torsion free semiprime ring and U a non-zero ideal of R .R admitting a non-zero derivation d satisfying d ([d(x),d(y)])=[x,y] for all x,y∈ U. If d acts as a homomorphism, then R contains a non-zero central ideal. Our aim in this paper is to investigate skew-commuting d and g be derivations on noncommutative prime ring and semiprime ring R.

Preliminaries
Throughout R will represent an associative ring with identity, Z(R) denoted to the center of R ,R is said to be n-torsion free, where n ≠ 0 is an integer, if whenever n x= 0,with x ∈ R, then x = 0. We recall that R is semiprime if xRx = (0) implies x = 0 and it is prime if xRy = (0) implies x = 0 or y = 0.A prime ring is semiprime but the converse is not true in general. An additive mapping d: R→R is called a derivation if d(xy) = d(x)y+ xd(y) holds for all x ,y∈R , and is said to be n-centralizing on U (resp. n-commuting on U ) , if  [21].
First we list the lemmas which will be needed in the sequel.

Lemma 3[14: Theorem 2]
Let R be a 2-torsion free semiprimering. If an additive mapping g:d→d is skew-commuting on R, then d=0.

Lemma 4 [15: Lemma 4]
Let R be a semiprimering and U a non-zero ideal of R .If d is a derivation of R which is centralizing on U, then d is commuting on U.

Theorem 3.1
Let R be a noncommutative prime ring, d and g be a for all x∈R, then d(R)=0 (resp. g(R)=0) or wd (resp. wg) is central for all w∈Z(R).
Proof: At first we suppose there exists an element say w∈ R, such that w ∈ Z(R). Let w be a non-zero element of Z(R).By linearizing our relation d(x)x+xg(x) ∈Z(R), we obtain d(x)y+d(y)x+xg(y)+yg(x) ∈Z(R) for all x,y∈R. (1) (2) Again in (1) replacing y by w 2 , we obtain d(x)w 2 +d(w 2 )x+xg(w 2 )+w 2 g(x) ∈ Z(R) for all x ∈ R, since d and g be a derivations of R and w ∈Z(R ), we obtain for all x∈R, then for all x∈R. (4) According to (2) the relation (4) gives for all x,y ∈ R.
Since w∈Z(R), then (5) gives Also from (2), we obtain ,y] for all x,y ∈ R. (7) Now from (6) and (7), we obtain w[d(x)w+wg(x),y]=0 for all x,y∈R. Since w∈Z(R ), this relation gives Replacing y by zy, with using (8), we get Replacing z by [d(x)+g(x),y]z and since R is prime ring , which implies Since w∈Z(R) and d,g are derivations, therefore, wd, wg and w(d+g) are derivations of R.

Theorem 3.2
Let R be a noncommutative prime ring, d be a skew-centralizing derivation of R (resp. g be a skewcentralizing derivation of R), if R admits to satisfy d(x)x+xg(x) ∈ Z(R) for all x ∈ R. Then d(R)=0 (resp. g(R)=0).
In particular, c(d(x o )x o +x o g(x o ))=c 2 =o. Since a semiprime ring has no nonzero central nilpotent, therefore, we get c=o, which implies d( If we taking d(x)=g(x), then d(x)x+xd(x)=0 for all x∈ R. Then by using Lemma 3, we obtain d(R)=0 (resp. g(R)=0).
If d(x)≠g(x),this case lead to d(x)x+xg(x) ∈Z(R) for all x ∈R. By Theorem3.1, we complete our proof.

Theorem 3.3
Let R be a 2-torsion free semiprime ring with cancellation property. If R admits a derivation d to satisfy (i) d acts as a skew-commuting on R.
(ii) d acts as a skew-centralizing on R . Then d(R) is commuting of R.
A according to (20) ,a above equation become d(xy)+d(yx)=o for all x,y∈ R.Then d(x)y+xd(y)+d(y)x+yd(x)=o for all x,y ∈ R .
Replacing y by x 2 and according to (20),we arrived to Then x2d(x)=-d(x)x2 for all x ∈ R.
By substituting (21) in (19) The proof of (ii) is similar. We complete the proof of theorem.

Remark 3.5
In Theorem 3.3 and Theorem 3.4, we can't exclude the condition char.R≠2, as it is shown in the following example.  Also when we have d acts as homomorphism (resp. acts as an anti-homomorphism).