Proof. Assume either d(I) 6= 0 or g(I) 6= 0. Thus, by Theorem 1.9 respectively we have that either [d(R), I] = (0) or [g(R), I] = (0). In any case, again by [13], R must contain some non-zero central ideals. Corollary 1.13. Let R be a 2-torsion free **semiprime** **ring** and F, G two gen- eralized derivations of R. If F ([x, y]) = [y, G(x)] for all x, y ∈ R, then ei- ther R contains a non-zero central ideal or there exist λ ∈ Z(R) such that F (x) = G(x) = λx, for all x ∈ R.

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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, proved that, let R be a **semiprime** **ring** and d a derivation of R with d 3 ≠0 .If [d

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In this work, we ﬁrst take into account the functional inequality which expands the func- tional inequality in []. It is well known that every **ring** left derivation (resp., **ring** left Jor- dan derivation) on a **semiprime** **ring** maps into its center; see [, ]. Considering the base of the previous result, we show that every approximate **ring** left derivation on a **semiprime** normed algebra maps into its center and then, by using this fact, we prove that every ap- proximate linear left derivation on a semisimple Banach algebra is continuous. We also establish the functional inequalities related to a linear derivation and their stability. In particular, mappings satisfying such functional inequalities on a **semiprime** Banach alge- bra are linear derivations which map into the intersection of the center and the radical. We ﬁnally investigate a linear generalized left Jordan derivation on a semisimple Banach algebra with application.

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Borut Zalar [12] worked on centralizers of **semiprime** rings and proved that Jordan centralizers and centralizers of this rings coincide. Joso Vukman[9,10,11] developed some remarkable results using centralizers on prime and **semiprime** rings. Vukman and Irena [8] proved that if R is a 2-tortion free **semiprime** **ring** and T : R → R is an additive mapping such that 2 T ( xyx ) = T ( x ) yx + xyx holds for all x , y ∈ R , then T is a centralizer.

On one hand, motivated by the work of Laradji and Thaheem 9, Vukman and Kosi- Ulbl 10, and Vukman 11, 12 on dependent elements of mappings of **semiprime** rings and on the other hand by the work done by various researchers on commuting derivations on prime and **semiprime** rings, we investigate some properties, not already investigated, of dependent elements of commuting derivations on **semiprime** rings. We show that the dependent elements of a commuting derivation of a **semiprime** **ring** are central and form a commutative **semiprime** subring of R. We also show that for a commuting derivation d on a **semiprime** **ring** R, there exist ideals U and V of R such that U ⊕ V is an essential ideal of R, U ∩ V {0}, d 0 on U, dV ⊆ V , and zero is the only dependent element of d \ V , the restriction of d on V ; that is, d acts freely on V .

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Every **ring** left derivation resp., **ring** derivation on **ring** is a Jordan left derivation resp., **ring** Jordan derivation. The converse is in general not true. It was shown by Ashraf and Rehman 20 that a **ring** Jordan left derivation on a 2-torsion free prime **ring** is a left derivation. In particular, a famous result due to Herstein 21 states that a **ring** Jordan derivation on a 2- torsion free **semiprime** **ring** is a derivation. In view of Thomas’ result 11, derivations on Banach algebras now belong to the noncommutative setting. Among various noncommutative versions of the Singer-Wermer theorem, Breˇsar and Vukman 22 proved the followings: every **ring** left derivation on a **semiprime** **ring** is derivation which maps into its center and also every continuous linear left derivation on a Banach algebra maps into its Jacobson radical.

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for all x, y ∈ I. Right multiplying (11) by y and then subtracting from (12), we have F(x)[F(y), y] = 0 for all x, y ∈ I. Putting xz for x in the last relation, we obtain F (x)z[F (y), y] = 0 for all x, y, z ∈ I. This implies [F (x), x]z[F (x), x] = 0 for all x, z ∈ I, that is, [F (x), x]I[F(x), x] = (0) for all x ∈ I. Since R is prime, it follows that [F(x), x] = 0 for all x ∈ I. Then by Lemma 2.5, F (x) = λx + ζ(x) for all x ∈ I, where λ ∈ C and ζ : I → C. Since F is additive, ζ is also additive map. Moreover, since F is left multiplier, so the map ζ(x) = F(x) − λx is also a left multiplier on I. This implies that F(x) = λx + ζ(x) for all x ∈ I. Hence, we obtain our conclusion (1). Theorem 3.4. Let R be a **semiprime** **ring**, I a nonzero ideal of R, and let F, d : R → R be two additive mappings such that F(xy) = F (x)y + xd(y) for all x, y ∈ R. If F (x)F (y) ± yx ∈ Z (R) for all x, y ∈ I, then [d(x), x] = 0 for all x ∈ I. Moreover, if d is a derivation such that d(I) ̸ = (0), then R contains a nonzero central ideal.

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Let R be a **ring** with center Z and I a nonzero ideal of R. An additive mapping F : R → R is called a generalized derivation of R if there exists a derivation d : R → R such that Fxy Fxy xdy for all x, y ∈ R. In the present paper, we prove that if Fx, y ±x, y for all x, y ∈ I or Fx ◦ y ±x ◦ y for all x, y ∈ I , then the **semiprime** **ring** R must contains a nonzero central ideal, provided d I / 0. In case R is prime **ring**, R must be commutative, provided d / 0. The cases i Fx, y ± x, y ∈ Z and ii Fx ◦ y ± x ◦ y ∈ Z for all x, y ∈ I are also studied.

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We study certain properties of derivations on **semiprime** rings. The main purpose is to prove the following result: let R be a **semiprime** **ring** with center Z(R), and let f , g be derivations of R such that f (x)x + xg(x) ∈ Z(R) for all x ∈ R, then f and g are central. As an application, we show that noncommutative semisimple Banach algebras do not admit nonzero linear derivations satisfying the above central property. We also show that every skew-centralizing derivation f of a **semiprime** **ring** R is skew-commuting.

It is also well know that the extended centroid of a prime **ring** is a field, however, for a **semiprime** **ring**, we can only assert that said extended centroid is a von Neumann regular **ring**. This is the cause of the difficulty of extending this result. The starting point of this path relies on the fact that each subset S of Q R s ( ) has an associated idempotent e [ ] S of the extended centroid C (see [4], Theorem 2.3.9) and on a consequence (see [4], Theorem 2.3.3 and Proposition 1.1 below) of the Weak Density Theorem ([4], Theorem 1.1.5).

P ROOF : Since any derivation d can be uniquely extended to a derivation in U , and U and R satisfy the same differential identities ( see [20]), then ad(x) ∈ Z(U ), for all x ∈ U . Let P ∈ M(C) such that U/P U is 2-torsion free. Note that U is also 2-torsion free **semiprime** **ring**. P U is a prime ideal of U , which is d−invariant by Lemma 3.1 denote U = U/P U and d the derivation induced by d on U . For any x ∈ U , ad(x) ∈ Z(U ). Since Z(U ) = (C + P U )/P U = C/P U , ad(x) ∈ (C + P U )/P U = C/P U, for all x ∈ U . Moreover U is a prime **ring**. Then it is clear that either d = 0; or a = 0 in U ; or [U , U ] = 0. This implies that, for any P ∈ M (C), either d(U ) ⊆ P U; or a ∈ P U ; or [U, U] ⊆ P U . In any case ad(U )[U, U ] ⊆ P U for any P ∈ M(C). By Lemma 3.1, T

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We investigate identities with derivations and automorphisms on **semiprime** rings. We prove, for example, that in case there exist a derivation D : R → R and an automorphism α : R → R, where R is a 2-torsion-free **semiprime** **ring**, such that [D(x)x + xα(x),x] = 0 holds for all x ∈ R, then D and α − I , where I denotes the identity mapping, map R into its center. Throughout this paper, R will represent an associative **ring** with center Z(R). A **ring** R is 2-torsion-free in case 2x = 0 implies that x = 0 for any x ∈ R. As usual, we write [x, y] for xy − yx and make use of the commutator identities [xy, z] = [x,z]y + x[y,z], [x, yz] = [x, y]z + y[x,z], x, y,z ∈ R. We denote by I the identity mapping of a **ring** R. An additive mapping D : R → R is called a derivation if D(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ R. Let α be an automorphism of a **ring** R. An additive map- ping D : R → R is called an α-derivation if D(xy ) = D(x)α(y) + xD(y) holds for all pairs x, y ∈ R. Note that the mapping D = α − I is an α-derivation. Of course, the concept of α-derivation generalizes the concept of derivation, since I -derivation is a derivation. We denote by C the extended centroid of a **semiprime** **ring** R and by Q Martindale **ring** of quotients. For the explanation of the extended centroid of a **semiprime** **ring** R and the Martindale **ring** of quotients, we refer the reader to [1]. A mapping f of R into itself is called centralizing on R if [ f (x), x] ∈ Z(R) holds for all x ∈ R; in the special case when [ f (x),x] = 0 holds for all x ∈ R, the mapping f is said to be commuting on R. The history of commuting and centralizing mappings goes back to 1955 when Divinsky [5] proved that a simple Artinian **ring** is commutative if it has a commuting nontrivial automor- phism. Two years later, Posner [8] has proved that the existence of a nonzero centralizing derivation on a prime **ring** forces the **ring** to be commutative (Posner’s second theorem). Luh [6] generalized the Divinsky result, we have just mentioned above, to arbitrary prime

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Theorem.3.2. Let n ≥2 be any fixed positive integer. Let R be non commutative n! – torsion free **semiprime** **ring** and I be any non – zero two sided ideal of R. Suppose that there exist a symmetric skew reverse n-derivation Ä: R n →R associated with an antiautomorphism á * . Let ä denote the trace of Ä such that ä is commuting on I and [ä(x),á * (x)] Z(R), for all x I then [ä(x), á * (x)] =0, x I. Proof. Our supposition is that there exist a symmetric skew reverse n – derivation Ä : R n → R associated with an antiautomorphism á * such that [ ä(x), x] = 0, for all x I and [ä(x), á * (x)]Z(R), xI.

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Semirings has been formally introduced by Vandiver in 1934. Golan [9] discussed the notion of semirings and their applications. In [4], Chandramouleeswaran and Tiruveni worked on the derivations on semirings. Zalar [1] studied centralizers on **semiprime** rings and proved that Jordan centralizer and centralizers of this rings coincide. In [15], Vukman and Irena proved that if ܴ is a 2-torsion free **semiprime** **ring** and ܶ: ܴ → ܴ is an additive mapping such that 2ܶ(ݔݕݔ) = ܶ(ݔ)ݕݔ + ݔݕܶ(ݔ) holds for all ݔ, ݕ ∈ ܴ, then ܶ is a centralizer. In papers [6,7,8] the authors Hoque and Paul worked on centralizers on **semiprime** Gamma rings and developed the results of [15] in Gamma rings. Motivated by this Florence and Murugesan [10] studied the notion of semirings and proved that Jordan centralizer of a 2-torsion free **semiprime** semiring is a centralizer. Here we develop the results of [7,15] in semirings by assuming that ܵ be a 2-torsion free **semiprime** semiring and ܶ ∶ ܵ → ܵ be an additive mappingsuch that 2ܶ(ݔݕݔ) = ܶ(ݔ)ݕݔ + ݔݕܶ(ݔ) holds for all ݔ, ݕ ∈ ܵ. Then ܶ is a centralizer.

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Further M stands for a prime Γ-**ring** with center ZM. The **ring** M is n-torsion-free if nx 0, x ∈ M implies x 0, where n is a positive integer, M is prime if aΓMΓb 0 implies a 0 or b 0, and it is **semiprime** if aΓMΓa 0 implies a 0. An additive mapping T : M → M is called a left right centralizer if T xαy T xαyT xαy xαTy for x, y ∈ M, α ∈ Γ and it is called a Jordan left right centralizer if

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Qualitative In this paper the terms d-system, m-system, n-system, U-ternary semigroup are introduced. It is proved that an ideal A of a ternary semigroup T is a prime ideal of T if and only if T\A is an m-system of T or empty. It is proved that an ideal A of a ternary semigroup T is completely **semiprime** if and only if T\A is a d-system of T or empty. It is proved that every m- system in a ternary semigroup T is an n-system. Further it is proved that an ideal Q of a ternary semigroup T is a **semiprime** ideal if and only if T\Q is an n-system of T (or) empty. It is proved that if N is an n-system in a ternary semigroup T and a N, then there exist an m-system M in T such that a M and M N. It is proved that a ternary semigroup T is U-ternary semigroup if and only if every ideal A of T is **semiprime** ideal of T. Further it is proved that if T is a ternary semigroup and A is an ideal of T, then T is U-ternary semigroup if and only if T\A is an n-system of T or empty and if T is U-ternary semigroup and A is an ideal of T, then T\A is an m-system of T.

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There is another approach stated by Kurzweg U H [15]. Seeing from his series of Blogs, one can see that Kurzweg actually tried to find out φ ( ) N to factorize the **semiprime** N . The idea was early seen in chapter 6 of YAN’s book [1]. Ac- tually, it is quite occasional for Kurzweg’s to factorize the numbers he reported because he has not brought an approach that finds out φ ( ) N inevitably.

A new class of rings, the class of left localizable rings, is introduced. A **ring** R is left localizable if each nonzero element of R is invertible in some left localization S −1 R of the **ring** R . Explicit criteria are given for a **ring** to be a left localizable **ring** provided the **ring** has only finitely many maximal left denominator sets (eg, this is the case if a **ring** has a left Artinian left quotient **ring**). It is proved that a **ring** with finitely many maximal left denominator sets is a left localizable **ring** iff its left quotient **ring** is a direct product of finitely many division rings. A characterization is given of the class of rings that are finite direct product of left localization maximal rings.

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In this section review of results on reverse derivations were presented. The reverse derivations on semi prime rings have been studied by Samman and Alyamani [9]. The authors obtain some characterizations of semi prime rings by using reverse derivations. The commutatively properties of a gamma **ring** with the derivations investigated [9] .The notion of orthogonality for two derivations on a semi prime **ring** initiated by Bresar and Vukman [6]. Some necessary and sufficient conditions for two orthogonal derivations to be are obtained. They also obtained a counterpart of a result of Posner [8]. Argac, on orthogonal generalized derivations on a semi prime **ring** and they established some results concerning two generalized derivations on a semi prime **ring** are worked by Nakajima and Albas [1] Ozturk, Jun and Kim [7] worked on prime - rings by means of derivations.

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Theorem 3.8 : Let R be a 2-torsion free **semiprime** *- **ring** , then every symmetric right Jordan * - Bicentralizer of R is a symmetric right reverse * - Bicentralizer . Where is an automorphism on R Lemma 3.9 : Let R be a *- **ring** with identity . Then a symmetric biadditive mapping T : R x R R is a symmetric left ( right ) reverse * - Bicentralizer if and only if T is of the form T(x , y) = a y * ) x * ) (T(x , y) = x * ) y * ) a ) for some fixed element a . Where is an automorphism on R .