SOME RESULTS ON SEMIGROUP IDEALS IN PRIME RING WITH DERIVATIONS

The purpose of the present paper is to prove some commutativity theorems in the setting of a semigroup ideal of a 3-prime near ring admitting a permuting generalized 3-derivation, thereby extending some known results of derivations, biderivations and 3-derivations.

Aequationes mathematicae

2016

Abstract. In the present paper, we investigate the commutativity of addition and ring behavior of 3-prime near-rings satisfying certain condi-tions involving generalized n-derivations on semigroup ideals. Moreover, examples justifying the necessity of the 3-primeness condition in all the results are provided. 1.

Ukrainian Mathematical Journal, 2012

Let R be a prime ring of characteristic not 2 and let I be a nonzero right ideal of R. Let U be the right Utumi quotient ring of R and let C be the center of U. If G is a generalized derivation of R such that [[G(x), x], G(x)] = 0 for all x ∈ I, then R is commutative or there exist a, b ∈ U such that G(x) = ax + xb for all x ∈ R and one of the following assertions is true: (1) (a − λ)I = (0) = (b + λ)I for some λ ∈ C, (2) (a − λ)I = (0) for some λ ∈ C and b ∈ C. Нехай R-просте кiльце, характеристика якого не дорiвнює 2, а I-ненульовий правий iдеал R. Нехай Uправе фактор-кiльце Утумi кiльця R, а C-центр U. Якщо G є узагальненим диференцiюванням R таким, що [[G(x), x], G(x)] = 0 для всiх x ∈ I, то R є комутативним або iснують a, b ∈ U такi, що G(x) = ax + xb для всiх x ∈ R i виконується одне з наступних тверджень: (1) (a − λ)I = (0) = (b + λ)I для деякого λ ∈ C, (2) (a − λ)I = (0) для деяких λ ∈ C та b ∈ C. 1. Introduction. Throughout this paper R will always denote a prime ring with center Z(R), extended centroid C, right Utumi quotient ring U (sometimes, as in [2], U is called the maximal right ring of quotients), and two-sided Martindale quotient ring Q (see [2] for the definitions). For any x, y ∈ R, the commutator of x and y is denoted by [x, y] and defined to be xy − yx. An additive mapping d from R into itself is called a derivation of R if d(xy) = d(x)y + xd(y) holds for all x, y ∈ R. An additive mapping g : R → R is called a generalized derivation of R if there exists a derivation d of R such that g(xy) = g(x)y + xd(y) for all x, y ∈ R [10]. Obviously any derivation is a generalized derivation. Moreover, other basic examples of generalized derivations are the mappings of the form x → ax + xb, for a, b ∈ R. A generalized derivation in this form is called (generalized) inner. Many authors have studied generalized derivations in the context of prime and semiprime rings (see [1, 10, 13, 14]). In [13], T. K. Lee extended the definition of a generalized derivation as follows. By a generalized derivation he means an additive mapping g : I → U such that g(xy) = g(x)y +xd(y) for all x, y ∈ I, where I is a dense right ideal of the prime ring R and d is a derivation from I into U. He also proved that every generalized derivation can be uniquely extended to a generalized derivation of U, and moreover, there exist a ∈ U and a derivation d of U such that g(x) = ax + d(x) for all x ∈ U [13] (Theorem 3). In [7], De Filippis proved that if R is a prime ring of characteristic not 2 and G is a generalized derivation of R such that [[G(x), x], G(x)] = 0 for all x ∈ R, then either R is commutative or there exists λ ∈ C such that G(x) = λx for all x ∈ R. In the same paper, he uses his result to prove a theorem concerning noncommutative Banach algebras. More precisely, he proves the following:

Journal of Taibah University for Science, 2015

A non-empty subset U of a near-ring N is said to be a semigroup left (resp. right) ideal of N if NU ⊆ U (resp. UN ⊆ U) and if U is both a semigroup left ideal and a semigroup right ideal, it will be called a semigroup ideal. In the present paper, we investigate the commutativity of addition and multiplication of near-rings satisfying certain identities involving n-derivations on semigroup ideals and ideals. Furthermore, we study the conditions with semigroup ideals for n-derivations D 1 and D 2 of N which imply that D 1 = D 2 .

The purpose of the present paper is to obtain the commutativity of a prime near ring N with a generalized derivation F associated with a nonzero derivation d satisfying one of the conditions: (i) [F (x), y] = ±y p (x • y)y q , (ii) [x, F (y)] = ±x p (x • y)x q , (iii) F (x) • y = ±y p [x, y]y q , (iv) x•F (y) = ±x p [x, y]x q , (v) F (x)•y = ±y p (x•y)y q , (vi) [x, F (y)] = ±x p [x, y]x q , (vii) [F (x), y] = ±y p [x, y]y q and (viii) x • F (y) = ±x p (x • y)x q for all x, y ∈ U , a semigroup ideal of N and p ≥ 0, q ≥ 0 are non-negative integers. Moreover, example proving the necessity of primeness hypothesis is given.

Algebra Colloquium, 2011

Let K be a commutative ring with unit, R be a prime K-algebra with center Z(R), right Utumi quotient ring U and extended centroid C, and I a nonzero right ideal of R. Let g be a nonzero generalized derivation of R and f(X1,…,Xn) a multilinear polynomial over K. If g(f(x1,…,xn)) f(x1,…,xn) ∈ C for all x1,…,xn ∈ I, then either f(x1,…,xn)xn+1 is an identity for I, or char (R)=2 and R satisfies the standard identity s4(x1,…,x4), unless when g(x)=ax+[x,b] for suitable a, b ∈ U and one of the following holds: (i) a, b ∈ C and f(x1,…,xn)2 is central valued on R; (ii) a ∈ C and f(x1,…,xn) is central valued on R; (iii) aI=0 and [f(x1,…,xn), xn+1]xn+2 is an identity for I; (iv) aI=0 and (b-β)I=0 for some β ∈ C.

International Journal of Mathematics and Mathematical Sciences, 2004

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.

2012

Let R be a ring with centre Z(R). An additive mapping F : R −→ R is said to be a generalized derivation if there exists a derivation d : R −→ R such that F(xy )= F(x)y+xd(y), for all x,y ∈ R (the map d is called the derivation associated with F). In the present note we prove that if a semiprime ring R admits a generalized derivation F, d is the nonzero associated derivation of F, satisfying certain polynomial constraints on a nonzero ideal I, then R contains a nonzero central ideal. Mathematics Subject Classification: Primary 16N60; Secondary 16W25

International Journal of Mathematics Trends and Technology, 2017

Let R be a prime ring and I be a non zero ideal of R. Suppose that F, G, H : R → R are generalized derivations associated with derivations d, g, h respectively. If the following holds (i)F (xy)+G(x)H(y)+[α(x), y] = 0; for all x, y ∈ I, where α is any map on R, then R is commutative.