# Semiprime ring

## Top PDF Semiprime ring:

### Lie Ideals and Generalized Derivations in Semiprime Rings

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.

### Skew Commuting Derivations of Noncommutative Prime Rings

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

### Asymptotic aspect of derivations in Banach algebras

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.

### An Equation Related to Centralizers in Semiprime Gamma Rings

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.

### Dependent Elements of Derivations on Semiprime Rings

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 .

### On Functional Inequalities Originating from Module Jordan Left Derivations

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.

### On Prime and Semiprime Rings with Additive Mappings and Derivations

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.

### Remarks on Generalized Derivations in Prime and Semiprime Rings

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.

### A note on a pair of derivations of semiprime rings

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.

### On Finite Rank Operators on Centrally Closed Semiprime Rings

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).

### Annihilator condition of a pair of derivations in prime and semiprime rings

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

### Identities with derivations and automorphisms on semiprime rings

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

### Symmetric Skew Reverse n-Derivations on Prime Rings and Semiprime rings

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),  xI.

### Some Relations Related to Centralizers on Semiprime Semiring

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.

### Generalized Derivations in Semiprime Gamma Rings

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

### A study on d system, m system and n system in ternary semigroups

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.

### A Parallel Probabilistic Approach to Factorize a Semiprime

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.

### Left localizable rings and their characterizations

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.