Prime Ring

Let R be a ring and α, β be automorphisms of R. An additive mapping F : R → R is called a generalized (α, β)-derivation on R if there exists an (α, β)derivation d: R → R such that F (xy) = F (x)α(y) + β(x)d(y) holds for all x, y ∈ R. For any x, y ∈ R, set [x, y] α,β = xα(y) − β(y)x and (x • y) α,β = xα(y) + β(y)x. In the present paper, we shall discuss the commutativity of a prime ring R admitting generalized (α, β)-derivations F and G satisfying any one of the following properties: (i) F ([x, y]) = (x • y) α,β , (ii) F (x • y) = [x, y] α,β , (iii) [F (x), y] α,β = (F (x) • y) α,β , (iv) F ([x, y]) = [F (x), y] α,β , (v) F (x • y) = (F (x) • y) α,β , (vi) F ([x, y] = [α(x), G(y)] and (vii) F (x • y) = (α(x) • G(y)) for all x, y in some appropriate subset of R. Finally, obtain some results on semi-projective Morita context with generalized (α, β)-derivations.

Let R be a ring with center Z(R), let n be a fixed positive integer, and let I be a nonzero ideal of R. A mapping h : x n ] = 0 respectively) for all x ∈ S. The following are proved: (1) if there exist generalized derivations F and G on an n!-torsion free semiprime ring R such that F 2 + G is n-commuting on R, then R contains a nonzero central ideal; (2) if there exist generalized derivations F and G on an n!-torsion free prime ring R such that F 2 + G is n-skew-commuting on I, then R is commutative.

Strongly prime rings may be defined as prime rings with simple central closure. This paper is concerned with further investigation of such rings. Various characterizations, particularly in terms of symmetric zero divisors, are given. We prove that the central closure of a strongly (semi-)prime ring may be obtained by a certain symmetric perfect one sided localization. Complements of strongly prime ideals are described in terms of strongly multiplicative sets of rings. Moreover, some relations between a ring and its multiplication ring are examined.

Let R be an associative ring. We define a subset S R of R as S R = {a ∈ R | aRa = (0)} and call it the source of semiprimeness of R. We first examine some basic properties of the subset S R in any ring R, and then define the notions such as R being a |S R |-reduced ring, a |S R |-domain and a |S R |-division ring which are slight generalizations of their classical versions. Beside others, we for instance prove that a finite |S R |-domain is necessarily unitary, and is in fact a |S R |-division ring. However, we provide an example showing that a finite |S R |-division ring does not need to be commutative. All possible values for characteristics of unitary |S R |-reduced rings and |S R |-domains are also determined.

The aim of this paper is to characterize those elements in a semiprime ring R for which taking local rings at elements and rings of quotients are commuting operations. If Q denotes the maximal ring of left quotients of R, then this happens precisely for those elements if R which are von Neumann regular in Q. An intrinsic characterization of such elements is given. We derive as a consequence that the maximal left quotient ring of a prime ring with a nonzero PI-element is primitive and has nonzero socle. If we change Q to the Martindale symmetric ring of quotients, or to the maximal symmetric ring of quotients of R, we obtain similar results: an element a in R is von Neumann regular if and only if the ring of quotients of the local ring of R at a is isomorphic to the local ring of Q at a.

In this paper, it is defined that left *-α-derivation, generalized left *-α-derivation and *-α-derivation, generalized *-α-derivation of a *-ring where α is a homomorphism. The results which proved for generalized left *-derivation of R in [1] are extended by using generalized left *-α-derivation. The commutativity of a *-ring with generalized left *-α-derivation is investigated and some results are given for generalized *-α-derivation.

In this paper we obtain some conditions which force prime rings to be primitive. Our main theorems are converses to well-known results on the primitivity of certain subrings of primitive rings. Applications are given to the case of primitive domains, and a tensor prod&t theorem is proved which answers a question of Herstein on the primitivity of E[x, ,..., x,J, for E the endomorphism ring of a vector space over a division ring.

In this paper, it is defined that left *-α-derivation, generalized left *-α-derivation and *-α-derivation, generalized *-α-derivation of a *-ring where α is a homomorphism. The results which proved for generalized left *-derivation of R in [1] are extended by using generalized left *-α-derivation. The commutativity of a *-ring with generalized left *-α-derivation is investigated and some results are given for generalized *-α-derivation.

Let R be a semiprime ring and F be a generalized derivation of R and n ≥ 1 a fixed integer. In this paper we prove the following: (1) If (F(xy) − yx)n is either zero or invertible for all ${x,y\in R}$ , then there exists a division ring D such that either R = D or R = M 2(D), the 2 × 2 matrix ring. (2) If R is a prime ring and I is a nonzero right ideal of R such that (F(xy) − yx)n  = 0 for all ${x,y \in I}$ , then [I, I]I = 0, F(x) = ax + xb for ${a,b\in R}$ and there exist ${\alpha, \beta \in C}$ , the extended centroid of R, such that (a − α)I = 0 and (b − β)I = 0, moreover ((a + b)x − x)I = 0 for all ${x\in I}$ .

In this paper, we generalize sone well-known commutativity theorems for associative rings as follows: Let ', > 1. ,,, .,, and be fixed nou-ncgative integers such that s ik m-1, or i/= n-1, and let R be a ring xvith unity satisfying the polynomial identity y*[x',y] [x,y']x for all y R. Sul,lose that (i) R has Q(z) (that is n[x,y] 0 implies [z,y] 0); (ii) the set of d] nilpotent ,,lem,'nts of R is central for > 0, and (iii) the set of all zero-divisors of R is also central hr > 0. Then R is commutative. If Q(n) is replaced by "rn and n are relatively prime pobitive integers," then R is commutative if extra constraint is given. Other related commutativity results are also obtained.

h t t p : / / j o u r n a l s. t u b i t a k. g o v. t r / m a t h / Abstract: Let R be a *-prime ring with characteristic not 2, σ, τ : R → R be two automorphisms, U be a nonzero *-(σ, τ)-Lie ideal of R such that τ commutes with * , and a, b be in R. (i) If a ∈ S * (R) and [U, a] = 0 , then a ∈ Z (R) or U ⊂ Z (R). (ii) If a ∈ S * (R) and [U, a] σ,τ ⊂ Cσ,τ , then a ∈ Z (R) or U ⊂ Z (R). (iii) If U ̸ ⊂ Z (R) and U ̸ ⊂ Cσ,τ , then there exists a nonzero *-ideal M of R such that [R, M ] σ,τ ⊂ U but [R, M ] σ,τ ̸ ⊂ Cσ,τ. (iv) Let U ̸ ⊂ Z (R) and U ̸ ⊂ Cσ,τ. If aU b = a * U b = 0 , then a = 0 or b = 0.

In this paper, it is defined that left *-α-derivation, generalized left *-α-derivation and *-α-derivation, generalized *-α-derivation of a *-ring where α is a homomorphism. The results which proved for generalized left *-derivation of R in [1] are extended by using generalized left *-α-derivation. The commutativity of a *-ring with generalized left *-α-derivation is investigated and some results are given for generalized *-α-derivation.

In this study, we prove that any nonzero reverse (,) − biderivation on a prime ring is (,) − biderivation. Also, we show that any Jordan (,) − biderivation on non-commutative semi-prime ring with ℎ () ≠ 2 is an (,) − biderivation. In addition, we investigate commutative feature of prime ring with Jordan left (,) − biderivation.

Abstract: Let R be a ∗ -prime ring with characteristic not 2, σ, τ : R → R be two automorphisms, U be a nonzero ∗ -(σ, τ)-Lie ideal of R such that τ commutes with ∗ , and a, b be in R. (i) If a ∈ S∗ (R) and [U, a] = 0, then a ∈ Z (R) or U ⊂ Z (R) . (ii) If a ∈ S∗ (R) and [U, a]σ,τ ⊂ Cσ,τ , then a ∈ Z (R) or U ⊂ Z (R) . (iii) If U ̸⊂ Z (R) and U ̸⊂ Cσ,τ , then there exists a nonzero ∗ -ideal M of R such that [R,M ]σ,τ ⊂ U but [R,M ]σ,τ ̸⊂ Cσ,τ . (iv) Let U ̸⊂ Z (R) and U ̸⊂ Cσ,τ . If aUb = a∗Ub = 0, then a = 0 or b = 0.

LetRbe a ring andSa nonempty subset ofR. Suppose thatθandϕare endomorphisms ofR. An additive mappingδ:R→Ris called a left(θ,ϕ)-derivation (resp., Jordan left(θ,ϕ)-derivation) onSifδ(xy)=θ(x)δ(y)+ϕ(y)δ(x)(resp.,δ(x2)=θ(x)δ(x)+ϕ(x)δ(x)) holds for allx,y∈S. Suppose thatJis a Jordan ideal and a subring of a2-torsion-free prime ringR. In the present paper, it is shown that ifθis an automorphism ofRsuch thatδ(x2)=2θ(x)δ(x)holds for allx∈J, then eitherJ⫅Z(R)orδ(J)=(0). Further, a study of left(θ,θ)-derivations of a prime ringRhas been made which acts either as a homomorphism or as an antihomomorphism of the ringR.

Let R be a 2-torsion free σ-prime ring with involution σ, U a nonzero Lie ideal of R and d : R −→ R a nonzero derivation commuting with σ. In this paper it is proved that if d 2 (U) = 0 then U ⊂ Z(R). Moreover, if charR = 3 and d 2 (U) ⊂ Z(R), then U ⊂ Z(R).

1. IntroductionLet R be an associative ring with center Z = Z(R). For each x;y 2 Rdenote the commutator xy yx by [x;y] and the anti-commutator xy +yxby xy. Recall that a ring R is prime if for any a;b 2 R, aRb = f0g impliesthat a = 0 or b = 0. An additive mapping d : R ! R is called a derivationif d(xy) = d(x)y +xd(y) for all x;y 2 R. In particular, for xed a 2 R, themapping I

Let R be a prime ring with characteristic di¨erent from two and U be a Lie ideal of R such that u 2 e U for all u e U. In the present paper it is shown that if d is an additive mappings of R into itself satisfying du 2 2udu, for all u e U, then either U t ZR or dU 0.

In this paper, we define a set including of all fa with a ∈ R generalized derivations of R and is denoted by f R. It is proved that (i) the mapping g : L (R) → f R given by g (a) = f −a for all a ∈ R is a Lie epimorphism with kernel Nσ,τ ; (ii) if R is a semiprime ring and σ is an epimorphism of R, the mapping h : f R → I (R) given by h (fa) = i σ(−a) is a Lie epimorphism with kernel l (f R) ; (iii) if f R is a prime Lie ring and A, B are Lie ideals of R, then [f A , f B ] = (0) implies that either f A = (0) or f B = (0).

Let R be a prime ring with characteristic di¨erent from two and U be a Lie ideal of R such that u 2 e U for all u e U. In the present paper it is shown that if d is an additive mappings of R into itself satisfying du 2 2udu, for all u e U, then either U t ZR or dU 0.

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.

The purpose of this paper is to establish some results concerning generalized left derivations in rings and Banach algebras. In fact, we prove the following results: Let R be a 2-torsion free semiprime ring, and let G : R −→ R be a generalized Jordan left derivation with associated Jordan left derivation δ : R −→ R. Then every generalized Jordan left derivation is a generalized left derivation on R. This result gives an affirmative answer to the question posed as a remark in Ashraf and Ali (Bull. Korean Math. Soc. 45:253-261, 2008). Also, the study of generalized left derivation has been made which acts as a homomorphism or as an anti-homomorphism on some appropriate subset of the ring R. Further, we introduce the notion of generalized left bi-derivation and prove that if a prime ring R admits a generalized left bi-derivation G with associated left bi-derivation B then either R is commutative or G is a right bi-centralizer(or bi-multiplier) on R. Finally, it is shown that every generalized Jordan left derivation on a semisimple Banach algebra is continuous.

Let A be a hereditary Noetherian prime ring that is not right primitive. A complete description of π-injective A-modules is obtained. Conditions under which the classical ring of quotients of A is a π-projective A-module are determined. A criterion for a right hereditary right Noetherian prime ring to be serial is obtained. Bibliography: 8 titles. All rings in this paper are assumed to be associative and with a non-zero identity element. Expressions such as 'a Noetherian ring' mean that the corresponding right-hand and left-hand conditions hold. The Jacobson radical of a module M is denoted by J(M). A module is hereditary if all its submodules are projective. A module is uniserial if any two of its submodules are comparable with respect to inclusion. A module M is serial if M is a direct sum of uniserial modules. A uniserial Noetherian domain A is complete if the ring A is complete with respect to the J(A)-adic topology. A module M is projective with respect to a module N (or N-projective) if for each epimorphism h: N → N and each homomorphism f : M → N there exists a homomorphism f : M → N such that hf = f. A module that is projective with respect to itself is called a quasiprojective (or self-projective) module. A module M is π-projective (see [1], p. 359) if for arbitrary submodules U and V of M such that U + V = M there exists an endomorphism f of M such that f(M) ⊆ U and (1 − f)(M) ⊆ V. If all factor modules of a module M are indecomposable, then M is clearly π-projective. Hence all uniserial modules are π-projective. Each quasiprojective module is π-projective [1], p. 359. Each quasicyclic Abelian group C(p ∞) is a π-projective non-quasiprojective module over the ring of integers. Recall that each Noetherian prime ring A has a simple Artinian classical ring of quotients Q, and Q A is an injective non-singular right (left) A-module. Conditions ensuring that the classical ring of quotients of a hereditary Noetherian prime, but not right primitive ring A is a quasiprojective right A-module are described in [2], Theorem 8. In this connection we present the following Theorems 1 and 2, which are the first two main results of this paper.

The concept of derivations as well as generalized derivations (i.e. I a,b (x) = ax + xb, for all a, b ∈ R) have been generalized as an additive function F : R −→ R satisfying F (xy) = F (x)y + xd(y) for all x, y ∈ R, where d is a nonzero derivation on R. Such a function F is said to be a generalized derivation. In the present paper it is shown that: if R is 2-torsion free prime ring, I = 0 an ideal of R and F a generalized derivation of R such that either F (xy) = F (x)F (y) or F (xy) = F (y)F (x) for all x, y ∈ I, then R is commutative.

In this paper, it is defined that left *-α-derivation, generalized left *-α-derivation and *-α-derivation, generalized *-α-derivation of a *-ring where α is a homomorphism. The results which proved for generalized left *-derivation of R in [1] are extended by using generalized left *-α-derivation. The commutativity of a *-ring with generalized left *-α-derivation is investigated and some results are given for generalized *-α-derivation.

Let R be a 2-torsion free prime ring. Suppose that ; are au- tomorphisms of R. In the present paper it is established that if R admits a nonzero Jordan left ( ; )-derivation, then R is commutative. Further, as an application of this resul it is shown that every Jordan left ( ; )-derivation on R is a left ( ; )-derivation on R. Finally, in case of an arbitrary prime ring it is proved that if R admits a left ( ; )-derivation which acts also as a homomorphism (resp. anti-homomorphism) on a nonzero ideal of R, then d = 0 on R.

h t t p : / / j o u r n a l s. t u b i t a k. g o v. t r / m a t h / Abstract: Let R be a *-prime ring with characteristic not 2, σ, τ : R → R be two automorphisms, U be a nonzero *-(σ, τ)-Lie ideal of R such that τ commutes with * , and a, b be in R. (i) If a ∈ S * (R) and [U, a] = 0 , then a ∈ Z (R) or U ⊂ Z (R). (ii) If a ∈ S * (R) and [U, a] σ,τ ⊂ Cσ,τ , then a ∈ Z (R) or U ⊂ Z (R). (iii) If U ̸ ⊂ Z (R) and U ̸ ⊂ Cσ,τ , then there exists a nonzero *-ideal M of R such that [R, M ] σ,τ ⊂ U but [R, M ] σ,τ ̸ ⊂ Cσ,τ. (iv) Let U ̸ ⊂ Z (R) and U ̸ ⊂ Cσ,τ. If aU b = a * U b = 0 , then a = 0 or b = 0.

Replacing invertibility with quasi-invertibility in Bass' first stable range condition we discover a new class of rings, the QB−rings. These constitute a considerable enlargement of the class of rings with stable rank one (B−rings), and include examples like End F (V ), the ring of endomorphisms of a vector space V over some field F, and B(F), the ring of all row-and column-finite matrices over F.

Let R be a prime ring that is not commutative and such that R ≇ M 2( GF (2)), let D, G be two generalized derivations of R, and let m, n be two fixed positive integers. Then D(xm)xn = xnG(xm) for all x ∈ R iff the following two conditions hold: (1) There exists w ∈ Q, the symmetric Martindale quotient ring of R, such that D(x) = xw and G(x) = wx for all x ∈ R; (2) either w ∈ C, or xm and xn are C-dependent for all x ∈ R. We also consider the situation for the semiprime case.

Let R be a ring and �,� be automorphisms of R. An additive mapping F: R → R is called a generalized (�,�)-derivation on R if there exists an (�,�)- derivation d: R → R such that F(xy) = F(x)�(y) + �(x)d(y) holds for all x,y ∈ R. For any x,y ∈ R, set (x,y)�,� = x�(y) − �(y)x and (x ◦ y)�,� = x�(y) + �(y)x. In the present paper, we shall discuss the commutativity of a prime ring R admitting generalized (�,�)-derivations F and G satisfying any one of the following properties: (i) F((x,y)) = (x ◦ y)�,�, (ii) F(x ◦ y) = (x,y)�,�, (iii) (F(x),y)�,� = (F(x) ◦ y)�,�, (iv) F((x,y)) = (F(x),y)�,�, (v) F(x ◦ y) = (F(x) ◦ y)�,�, (vi) F((x,y) = (�(x),G(y)) and (vii) F(x ◦ y) = (�(x) ◦ G(y)) for all x,y in some appropriate subset of R. Finally, obtain some results on semi-projective Morita context with generalized (�,�)-derivations.

Let R be a 2-torsion free prime ring. Suppose that ; are au- tomorphisms of R. In the present paper it is established that if R admits a nonzero Jordan left ( ; )-derivation, then R is commutative. Further, as an application of this resul it is shown that every Jordan left ( ; )-derivation on R is a left ( ; )-derivation on R. Finally, in case of an arbitrary prime ring it is proved that if R admits a left ( ; )-derivation which acts also as a homomorphism (resp. anti-homomorphism) on a nonzero ideal of R, then d = 0 on R.

Let R be a prime ring with characteristic di¨erent from two and U be a Lie ideal of R such that u 2 e U for all u e U. In the present paper it is shown that if d is an additive mappings of R into itself satisfying du 2 2udu, for all u e U, then either U t ZR or dU 0.

Let R be a 2-torsion free prime ring. Suppose that ; are au- tomorphisms of R. In the present paper it is established that if R admits a nonzero Jordan left ( ; )-derivation, then R is commutative. Further, as an application of this resul it is shown that every Jordan left ( ; )-derivation on R is a left ( ; )-derivation on R. Finally, in case of an arbitrary prime ring it is proved that if R admits a left ( ; )-derivation which acts also as a homomorphism (resp. anti-homomorphism) on a nonzero ideal of R, then d = 0 on R.

LetRbe a ring andSa nonempty subset ofR. Suppose thatθandϕare endomorphisms ofR. An additive mappingδ:R→Ris called a left(θ,ϕ)-derivation (resp., Jordan left(θ,ϕ)-derivation) onSifδ(xy)=θ(x)δ(y)+ϕ(y)δ(x)(resp.,δ(x2)=θ(x)δ(x)+ϕ(x)δ(x)) holds for allx,y∈S. Suppose thatJis a Jordan ideal and a subring of a2-torsion-free prime ringR. In the present paper, it is shown that ifθis an automorphism ofRsuch thatδ(x2)=2θ(x)δ(x)holds for allx∈J, then eitherJ⫅Z(R)orδ(J)=(0). Further, a study of left(θ,θ)-derivations of a prime ringRhas been made which acts either as a homomorphism or as an antihomomorphism of the ringR.

Let R be a prime ring with characteristic not two. U a (σ, τ)-left Lie ideal of R and d : R → R a non-zero derivation. The purpose of this paper is to invesitigate identities satisfied on prime rings. We prove the following results: (1) [d(R), a] = 0 ⇔ d([R, a]) = 0. (2) if (R, a) σ,τ = 0 then a ∈ Z. (3) if (R, a) σ,τ ⊂ C σ,τ then a ∈ Z. (4) if (U, a) ⊂ Z then a 2 ∈ Z or σ(u) + τ (u) ∈ Z, for all u ∈ U. (5) if (U, R) σ,τ ⊂ C σ,τ then U ⊂ Z.

In this paper the Lie structure of prime rings of characteristic 2 is discussed. Results on Lie ideals are obtained. These results are then applied to the group of units of the ring, and also to Lie ideals of the symmetric elements when the ring has an involution. This work extends recent results of I. N. Herstein, C. Lanski and T. S. Erickson on prime rings whose characteristic is not 2, and results of S. Montgomery on simple rings of characteristic 2. LEMMA 9. If U is a Lie ideal of R, xe R, x 2 = 0 and xUx = 0, eΐίfeer # = 0 or UaZ.