Symmetric Bi-derivation Research Articles

Commutativity theorems on Lie ideals with symmetric bi-derivations in semiprime rings

Let [Formula: see text] be a semiprime ring, [Formula: see text] a square-closed Lie ideal of [Formula: see text] and [Formula: see text] a symmetric bi-derivation and [Formula: see text] be the trace of [Formula: see text] In this paper, we shall prove that [Formula: see text] contains a nonzero central ideal if any one of the following holds: (i) [Formula: see text] (ii) [Formula: see text] (iii) [Formula: see text] (iv) [Formula: see text] (v) [Formula: see text] (vi) [Formula: see text] (vii) [Formula: see text] (viii) [Formula: see text] (ix) [Formula: see text] (x) [Formula: see text] (xi) [Formula: see text] and (xii) [Formula: see text], for all [Formula: see text]

Generalized symmetric bi-derivations in MV-algebras

In this study, we define the generalized symmetric bi-derivation and its trace on MV-algebras and give its examples. We investigate some properties of generalized symmetric bi-derivation’s trace. We introduce isotone generalized symmetric bi-derivation and bi-additive generalized symmetric bi-derivation. We show that if A is a linearly ordered MV-algebra, Γ is a bi-additive generalized symmetric bi-derivation on A related to symmetric bi-derivation D and γ is a trace of Γ , then γ = 0 or γ1 = 1

Symmetric bi-derivation on bitonic algebras

In this study, we give definition of symmetric bi-derivation on bitonic algebras and investigate its properties.

Symmetric bi-derivations of subtraction algebras

In this paper, we introduce the notion of symmetric bi-derivations on subtraction algebra and investigated some related properties. We prove that a map $D : X\times X\to X$ is a symmetric bi-derivation on $ X$ if and only if $D$ is a symmetric map and it satisfies $D(x-y, z)=D(x, z)-y$ for all $x, y, z\in X.$

Symmetric bi-derivations and their generalizations on group algebras

Here, we investigate symmetric bi-derivations and their generalizations on L? 0 (G)*. For k ? N, we show that if B:L?0(G)*x L?0(G)* ? L?0(G)* is asymmetric bi-derivation such that [B(m,m),mk] ? Z(L?0(G)*) for all m ? L? 0 (G)*, then B is the zero map. Furthermore, we characterize symmetric generalized biderivations on group algebras. We also prove that any symmetric Jordan bi-derivation on L? 0(G)* is a symmetric bi-derivation.

On symmetric Jordan and Jordan left bi-derivations of prime rings

In this paper, we investigated some properties of symmetric Jordan bi-derivation and symmetric Jordan left bi-derivation for associative rings. We showed that for an associative prime ring with $$charR\ne 2$$ if D is a symmetric Jordan bi-derivation then D is symmetric bi-derivation. And also we showed that for a 2-torsion free and 3-torsion free prime ring, if there exists a nonzero symmetric Jordan left bi-derivation D then R is commutative.

Commutatively of Prime and Semiprime ?-Rings with Symmetric BI-Derivations

Let M be a ?-ring and let D: M x M ->M be a symmetric bi-derivation with the trace d: M -> M denoted by d(x) = D(x, x) for all x?M. The objective of this paper is to prove some results concerning symmetric bi-derivation on prime and semiprime ?-rings. If M is a 2-torsion free prime ?-ring and D ? 0 be a symmetric bi-derivation with the trace d having the property d(x)?x - x?d(x) = 0 for all x?M and ???, then M is commutative. We also prove another result in ?-rings setting analogous to that of Posner for prime rings.GANIT J. Bangladesh Math. Soc.Vol. 34 (2014) 27-33

Symmetric Reverse Bi-Derivations on Prime Rings

Let R be a 2, 3-torsion free prime ring. Let D:(.,.):R×R?Rand dbe a symmetric reverse bi-derivation and the trace of D respectively.If d is commuting orcentralizing on R.Then D=0.LetD_1:(.,.):R×R?R,? D?_2:(.,.):R×R?R aresymmetric reverse bi-derivations and B(.,.):R×R?R be a symmetric bi-additive mapping. If D_1 (d_2 (x),x)=0 and d_1 (d_2 (x) )=f(x), for all x?R, where d_1,? d?_2and f are the traces ofD_1,? D?_2 and B. In this case either D_1=0or D_2=0.

ON SYMMETRIC BI-DERIVATIONS OF KU-ALGEBRAS

The notion of left-right (resp. right-left) symmetric biderivation of KU-algebras is introduced and some related properties are investigated.

SYMMETRIC BI-DERIVATIONS OF BCH-ALGEBRAS

Abstract. The aim of this paper is to introduce the notion of left-right (resp. right-left) symmetric bi-derivation of BCH-algebrasand some related properties are investigated. 1. IntroductionIn 1966, Imai and Iseki introduced two classes of abstract algebras,BCK-algebra and BCI-algebras [6]. It is known that the class of BCI-algebras is a generalization of the class of BCK-algebras In 1983, Hu andLi [3] introduced the notion of a BCH-algebra, which is a generalizationof the notions of BCK-algebras and BCI-algebras. They have studied afew properties of these algebras. In this paper, we introduce the notionof left-right (resp. right-left) symmetric bi-derivations of BCH algebrasand investigate some properties of symmetric bi-derivations in a BCH-algebra. Moreover, we prove that the set of all symmetric bi-derivationson a medial BCH-algebra forms a semigroup under a suitably de nedbinary composition.2. PreliminaryBy a BCH-algebra, we mean an algebra (X;;0) with a single binaryoperation that satis es the following identities for any x;y;z 2X :(BCH1) xx = 0,(BCH2) xy = 0 and y x = 0 imply x = y;(BCH3) (xy) z = (xz) y; where x y if and only if x*y=0.In a BCH-algebra, the following identities are true for all x;y 2X:

SYMMETRIC BI-DERIVATIONS ON TM-ALGEBRAS

Recently an algebra based on propositional calculi was introdued by Tamilarasi and Mekalai in the year 2010 known as TM−algebras, see (6). In our paper (1) we introduced the notion of derivation on TM-algebras. In this paper, we introduce the notion of symmetric bi-derivation on TM-algebras and study some of its properties.

Symmetric bi-derivations of lattices

In this paper, we introduced the notion of symmetric bi-derivation on lattices and investigated some related properties. We characterized the distributive and modular lattices by symmetric bi-derivations.

On a theorem of J. Vukman

LetR be a ring. A bi-additive symmetric mappingD:R × R → R is called a symmetric bi-derivation if, for any fixedy ∈ R, the mappingx ↦ D(x, y) is a derivation. J. Vukman [2, Theorem 2] proved that, ifR is a non-commutative prime ring of characteristic not two and three, and ifD:R × R → R is a symmetric bi-derivation such that [D(x, x), x] lies in the center ofR for allx ∈ R, thenD = 0. This result is in the spirit of the well-known theorem of Posner [1, Theorem 2], which states that the existence of a nonzero derivationd on a prime ringR, such that [d(x), x] lies in the center ofR for allx ∈ R, forcesR to be commutative. In this paper we generalize the result of J. Vukman mentioned above to nonzero two-sided ideals of prime rings of characteristic not two and we prove the following.

On a conjecture of Vukman

Let R be a ring A bi‐additive symmetric mapping d : R × R → R is called a symmetric bi‐derivation if, for any fixed y ∈ R, the mapping x → D(x, y) is a derivation. The purpose of this paper is to prove the following conjecture of Vukman.Let R be a noncommutative prime ring with suitable characteristic restrictions, and let D : R × R → R and f : x → D(x, x) be a symmetric bi‐derivation and its trace, respectively. Suppose that fn(x) ∈ Z(R) for all x ∈ R, where fk+1(x) = [fk(x), x] for k ≥ 1 and f1(x) = f(x), then D = 0.

Two results concerning symmetric bi-derivations on prime rings

LetR be a ring. A bi-additive symmetric mappingD(.,.): R × R → R is called a symmetric bi-derivation if, for any fixedy ∈ R, the mappingx → D(x, y) is a derivation. The purpose of this paper is to prove two results concerning symmetric bi-derivations on prime rings. The first result states that, ifD1 andD2 are symmetric bi-derivations on a prime ring of characteristic different from two and three such thatD1(x, x)D2(x,x) = 0 holds for allx ∈ R, then eitherD1 = 0 orD2 = 0. The second result proves that the existence of a nonzero symmetric bi-derivation on a prime ring of characteristic different from two and three, such that [[D(x, x),x],x] ∈ Z(R) holds for allx ∈ R, whereZ(R) denotes the center ofR, forcesR to be commutative.

Symmetric bi-derivations on prime and semi-prime rings

LetR be a ring. A bi-additive symmetric mappingD(.,.): R × R → R is called a symmetric bi-derivation if, for any fixedy ∈ R, a mappingx ↦ D(x, y) is a derivation. The purpose of this paper is to prove some results concerning symmetric bi-derivations on prime and semi-prime rings. We prove that the existence of a nonzero symmetric bi-derivationD(.,.): R × R → R, whereR is a prime ring of characteristic not two, with the propertyD(x, x)x = xD(x, x), x ∈ R, forcesR to be commutative. A theorem in the spirit of a classical result first proved by E. Posner, which states that, ifR is a prime ring of characteristic not two andD 1,D 2 are nonzero derivations onR, then the mappingx ↦ D 1(D 2 (x)) cannot be a derivation, is also presented.