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Some Results on Symmetric Reverse ∗-n-Derivations
Abstract
In this paper, the commuting and centralizing of symmetric reverse ∗- -derivation on Lie ideal are studied and the commutativity of prime ∗-ring with the concept of symmetric reverse ∗- -derivations are proved under certain conditions.
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In this paper we introduce the notation of symmetric skew 4-derivation of Semiprime ring and we consider R be a non-commutative 2, 3-torsion free semi prime ring, I be a non zero two sided ideal of R, a; be anautomorphism of R, and D:R4→ R be a symmetric skew 4-derivation associated with the automorphism a. If f is trace of D such that [f(x), a(x)] ∈ Z for all x∈I, then [f(x), a(x)] =0, for all x∈I.
Let be a fixed integer, R be a noncommutative n!-torsion free ring and I be any non zero ideal of R. In this paper we have proved the following results; (i) If R is a prime ring and there exists a symmetric skew n-derivation associated with the automorphism on R, such that the trace function of D satisfies , for all then (ii) If R is a semi prime ring and the trace function commuting on I, satisfies , for all then , for all Moreover, we have proved some annihilating conditions for algebraic identity involving multiplicative(generalized) derivation.
Let R be a *-ring. In this paper we introduce the notion of *-n-derivation in R. An additive mapping x -> x* of R into itself is called an involution on R if it satisfies the conditions: (i) (x*)* = x, (ii) (xy)* = y*x* for all x, y is an element of R. A ring R equipped with an involution '*' is called a *-ring. It is shown that if a prime *-ring R admits a nonzero *-n-derivation (resp. a reverse *-n-derivation) D, then R is commutative. Further, some related properties of *-n-derivation in a semiprime *-ring have also been investigated. Finally, a structure theorem for *-n-derivation has been obtained.
In this paper we introduce the notion of symmetric skew 3-derivations of prime or semiprime rings and prove that under certain conditions a prime ring with a nonzero symmetric skew 3-derivation has to be commutative.
Let R be a ring and X be a left R-module. The purpose of this paper is to investigate additive mappings D1: R → X and D2: R → X that satisfy D1(ab) = aD1(b) + bD1(a), a, b ∈ R (left derivation) and D2(a2) = 2aD2(a), a ∈ R (Jordan left derivation). We show, by the rather weak assumptions, that the existence of a nonzero Jordan left derivation of R into X implies R is commutative. This result is used to prove two noncommutative extensions of the classical Singer-Wermer theorem.
In this chapter we shall make a study of rings satisfying certain ascending chain conditions. In the non-commutative case-and
this is really the only case with which we shall be concerned- the decisive and incisive results are three theorems due to
Goldie. The main part of the chapter will be taken up with a presentation of these.
Definition. An element a in the ring R is regular if it is neither a left nor right zero divisor in R.
LetR be a *-ring. We study an additive mappingD: R R satisfyingD(x
2) =D(x)x
* +xD(x) for allx R.It is shown that, in caseR contains the unit element, the element 1/2, and an invertible skew-hermitian element which lies in the center ofR, then there exists ana R such thatD(x) = ax
*
– xa for allx R. IfR is a noncommutative prime real algebra, thenD is linear. In our main result we prove that a noncommutative prime ring of characteristic different from 2 is normal (i.e.xx
* =x
*
x for allx R) if and only if there exists a nonzero additive mappingD: R R satisfyingD(x
2) =D(x)x
* +xD(x) and [D(x), x] = 0 for allx R. This result is in the spirit of the well-known theorem of E. Posner, 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.
Jordan Permuting 3-Derivations of Prime Rings
- A K Faraj
- S J Shareef
A. K. Faraj, and S. J. Shareef, "Jordan Permuting 3-Derivations of Prime Rings", Iraqi Journal of Science, 58(2017),
No. 2A, 687-693.
On generalized * -n-Derivation in * -Ring
- U K Sharma
- S Kumar
U. K. Sharma, and S. Kumar, "On generalized * -n-Derivation in * -Ring", Global Journal of Pure and Applied
Math., 13(2017), No. 10, 7561-7572.