ON SOME EQUATIONS RELATED TO DERIVATIONS IN RINGS

Let m and n be positive integers with m + n ≠ 0 , and let R be an ( m + n + 2 ) ! -torsion free semiprime ring with identity element. Suppose there exists an additive mapping D : R → R , such that D ( x m + n + 1 ) = ( m + n + 1 ) x m D ( x ) x n is fulfilled for all x ∈ R , then D is a derivation which maps R into its center.

Theorem 2. For integers m, n with m ≥ 0, n ≥ 0, and m + n = 0, let R be an (m + n + 2)!torsion free semiprime ring with identity element.Suppose there exists an additive mapping D : R → R, such that D(x m+n+1 ) = (m + n + 1)x m D(x)x n is fulfilled for all x ∈ R. In this case, D is a derivation, which maps R into its center.In case R is a noncommutative prime ring, we have D = 0.
In case m = 1, n = 0 (we adopt the convention x 0 = e, for all x ∈ R, where e denotes the identity element), we have an additive mapping satisfying the relation D(x 2 ) = 2xD(x), x ∈ R. Such mappings are called left Jordan derivations (see [8,10,15]).Brešar and Vukman [8,Corollary1.3] have proved that the existence of a nonzero Jordan derivation on a 2-and 3-torsion free prime ring forces the ring to be commutative.For the proof of Theorem 2, we need Theorem 4, which is of independent interest.For the proof of Theorem 4 the lemma below will be needed.We refer the reader to [3] for the definitions and an account of the theory of the extended centroid and central closure as well as related topics and to [6] for an introductory survey on functional identities.Lemma 3. Let R be a 2-torsion free prime ring and let A be its central closure.Suppose that an additive mapping F : Proof.In the case when F maps into R, the lemma was first proved by Brešar in [5,Theorem 2].Fortunately, the same proof works in the case when F maps into A (on the other hand, see, e.g., [2] for a more general result).
Theorem 4. Let R be a 2-torsion free semiprime ring.Suppose that an additive mapping F : R → R satisfies [[F(x),x],x] = 0 for all x ∈ R.Then, [F(x),x] = 0 holds for all x ∈ R.
Proof.Since R is semiprime, there exists a family of prime ideals {P α ; α ∈ A} such that ∩ α P α = (0).Moreover, without loss of generality, we may assume that the prime rings R α = R/P α are 2-torsion free (see, e.g., [1, page 459]).Now fix some P = P α , α ∈ A. The theorem will be proved by showing that [F(x),x] ∈ P for every x ∈ R. Given x ∈ R, we will write x for the coset x + P ∈ R/P.By C, we denote the extended centroid of the prime ring R/P, and by A the central closure of R/P .One can consider A as a vector space over the field C. Since C can be regarded as a subspace of A, there exists a subspace B of A such that Lemma 2] for that matter), it follows that [F(p),x] = 0 for all x ∈ R, p ∈ P, that is, F(p) lies in the center of R/P.In particular, πF(p) = 0. Using this, we see that the mapping F : R/P → A, F(x) = πF(x) is well defined.Note that F is additive and satisfies [[F(x),x],x] = 0 for all x ∈ R.But then the lemma shows that [F(x),x] = 0 for all x ∈ R, which implies that [F(x),x] ∈ P. The proof of the theorem is complete.Theorem 4 generalizes Theorem 2 proved by Brešar [5] and Theorem 2 proved by Vukman in [14].

Proof of Theorem 2. From the relation
J. Vukman and I. Kosi-Ulbl 2705 it follows immediately that where e denotes the identity element.Putting x + e for x in the relation (1) and using (2), we obtain ( Using (1) and collecting together terms of (3) involving the same number of factors of e, we obtain where f i (x,e) stands for the expression of terms involving i factors of e. Replacing x by x + 2e,x + 3e,...,x + (m + n)e in turn in (1) and expressing the resulting system of m + n homogeneous equations, we see that the coefficient matrix of the system is a van der Monde matrix Since the determinant of the matrix is different from zero, it follows that the system has only a trivial solution.
In particular, Since R is a (m + n + 2)!-torsion free ring, the above equations reduce to respectively.We intend to prove that the mapping x → [D(x),x] is commuting on R. For this purpose, we write in x + y for x in (8), which gives Putting y = (m + n)x 2 in the relation above, we obtain According to (8), the above relation reduces to Subtracting ( 9) from ( 12), we obtain From the above relation, we obtain Subtracting ( 14) from (12), one obtains Since R is (m + n + 2)!-torsion free ring, the above relation reduces to which can be written in the form Now Theorem 4 makes it possible to conclude that J. Vukman and I. Kosi-Ulbl 2707 In other words, D is commuting on R. The fact that D is commuting on R makes it possible to replace D(x)x in (8) by xD(x).The relation (8) reduces to D(x 2 ) = 2xD(x), x ∈ R. Using again the fact that D is commuting, we obtain D(x 2 ) = D(x)x + xD(x), x ∈ R. In other words, D is a Jordan derivation.Let us recall that any Jordan derivation on a 2-torsion free semiprime ring is a derivation.It is well known and easy to prove that any commuting derivation on a semiprime ring R maps R into Z(R) (see [15]).In case R is a noncommutative prime ring, Posner's second theorem completes the proof of the theorem.
In the proof of Theorem 2, we met an additive mapping D satisfying the relation below In case n = 0 and R is an m-torsion free ring, we have an additive mapping D satisfying the relation D(x 2 ) = 2xD(x), x ∈ R. In other words, D is a left Jordan derivation.It was proved (see [15,Theorem 1]) that left Jordan derivations on a 2-and 3-torsion free semiprime ring are derivations which map the ring into its center.These observations lead to the conjecture.Conjecture 5. Let R be a semiprime ring with suitable torsion restrictions.Suppose there exists an additive mapping D : R → R satisfying the relation for all x ∈ R and some integers m ≥ 0, n ≥ 0, m + n = 0.In case m = n, the mapping D is a derivation which maps R into Z(R).
Our next result is related to the conjecture above.