Jordan Ideals Research Articles

Jordan ideals and $$(\alpha , \beta )$$-derivations on 3-prime near-rings and rings

Jordan ideals and $$(\alpha , \beta )$$-derivations on 3-prime near-rings and rings

On Jordan ideals with left derivations in 3-prime near-rings

We will extend in this paper some results about commutativity of Jordan ideals proved in [2] and [6]. However, we will consider left derivations instead of derivations, which is enough to get good results in relation to the structure of near-rings. We will also show that the conditions imposed in the paper cannot be removed.

On zero-power valued homoderivations in 3-prime near-rings

We will start this article by proving a crucial concept, which will allow us to overcome a set of obstacles we encountered in previous articles concerning the commutativity of near-ring involving homoderivations and Jordan ideals. Furthermore, we present examples to show that limitations imposed in the hypothesis of our results are necessary.

On Jordan ideals and Generalized (α, 1)- Reverse derivations in ∗-prime rings

Let R will be a 2- torsion free ∗-prime ring and α be an automorphisum of R. F be a nonzero generalized (α, 1)- reverse derivation of R with associated nonzero (α, 1)- reverse derivation d which commutes with ∗ and J be a nonzero ∗-Jordan ideal and a subring of R. In the present paper, we shall prove that R is commutative if any one of the following holds: (i) [F(u), u]α,1 = 0, (ii) F(u) α(u) = ud(u), (iii) F(u2) = ± α(u2), (iv) F(u2) = 2d(u) α(u), (v) d(u2) = 2F(u) α(u), for all u ∈ U.

On Generalized Semi-derivations and Jordan Ideals

On Generalized Semi-derivations and Jordan Ideals

Some Identities of 3-Prime Near-Rings Involving Jordan Ideals and Left Generalized Derivations

In the current paper, we study the structure of Jordan ideals of a 3-prime near-ring which satisfies some algebraic identities involving left generalized derivations and right centralizers. The limitations imposed in the hypothesis were justified by examples.

ON COMMUTATIVITY OF SEMIRINGS THROUGH IDENTITIES OF GENERALIZED DERIVATIONS

The main purpose of this paper is to investigate the commuting conditions for prime MA-semirings through Jordan ideals and generalized derivations which are responsible to extend a few results of rings.

Jordan and Lie ideals of inverse semirings

In this paper, we consider the notion of Jordan ideals and the Lie ideals in inverse semirings. It includes generalization of some well-known results of Herstein, in the setting of inverse semirings.

On differential identities of Jordan ideals of semirings

In this article some salient characteristics of Jordan ideals of MA-semirings are discussed. We prove some results for derivations of MA-semirings satisfying different identities on their Jordan ideals and investigate commuting conditions through these ideals.

On Posner’s second theorem for semirings with involution

EC Posner [14] proved a commutativity result known as Posner’s second theorem for a derivation centralizing on a prime ring. In this paper we prove some results for *-Jordan ideals of MA-semirings with involution. Main objective of this paper is to prove the aforesaid result for *-prime MA-semirings.

On generalized derivations and Jordan ideals of prime rings

Let R be a 2-torsion free prime ring with center Z(R) and a nonzero Jordan ideal J. In this paper, our main objective is to prove that: If $$F:R\rightarrow R$$ is a generalized derivation associated with a derivation d of R such that $$[F(u),u]_{k}\in Z(R)$$ for all $$u\in J$$ and a fixed integer $$k\ge 1,$$ then either R is commutative or there exists some $$a\in C$$ such that $$F(x)=ax$$ for all $$x\in R,$$ which extends a result of Soufi and Aboubakr (Turk. J. Math. 38:233–239, 2014, Theorem 3.2). Moreover, a Jordan ideal which is also a subring seems close to an ideal. Therefore, results on Jordan ideals (which are not necessarily subrings) are more interesting. We show that all the theorems of Soufi and Aboubakr [20] are true for Jordan ideals (which are not necessarily subrings).

On Jordan Ideals of Inverse Semirings with Involution

Objectives: The main objective of this article is to introduce *-Jordan ideals of a certain class of semirings called MA-semirings with involution and to investigate some conditions for which the above said ideals contained in the center. Methods and findings: We use the Jacobian identities and 2-torsion freeness of MA semirings. In this connection, we establish some important results of ring theory for the class of MA-semirings. Applications/ improvements: The commutative property is helpful to study the theory of semirings with ease therefore we find some conditions to impose commutativity in semirings, which are indeed novel idea in the field of semirings. Furthermore, these conditions are used in a most generalized way that these conditions bring the *-Jordan ideals to the center, therefore, it would be the corollary of result that semiring is commutative. Keywords: Semirings, *-Semirings, MA-Semirings, *-Jordan Ideals, *-Prime Semirings, Involution Mathematics Subject Classifications (2010): 16Y60, 16W10

Commutativity of a 3-Prime near Ring Satisfying Certain Differential Identities on Jordan Ideals

The purpose of this study is to obtain the commutativity of a 3-prime near ring satisfying some differential identities on Jordan ideals involving derivations and multiplicative derivations. Further, herein we discuss some examples to show the necessity of the hypothesis to our results.

Generalized Left Jordan ideals In Prime Rings

Let R be a prime ring and U be a (σ,τ)-left Jordan ideal .Then in this paper, we proved the following , if aU Z (Ua Z), a R, then a = 0 or U Z. If aU C s,t (Ua C s,t), a R, then either a = 0 or U Z. If 0 ≠ [U,U] s,t .Then U Z. If 0≠[U,U] s,t C s,t, then U Z .Also, we checked the converse some of these theorems and showed that are not true, so we give an example for them.

Correction: A note on derivations and Jordan ideals in prime rings

Correction: A note on derivations and Jordan ideals in prime rings

Some commutativity criteria for prime rings with differential identities on Jordan ideals

In this article, we present some commutativity theorems for a ring R equipped with a generalized derivation satisfying certain differential identities on Jordan ideals of R. Some related results for prime rings are also discussed. Finally; we provide examples to show that the assumed restrictions are not superfluous.

Commutativity of near-rings with certain constrains on Jordan ideals

The purpose of this paper is to study derivations satisfying certain differential identities on Jordan ideals of 3-prime near-rings. Moreover, we provide examples to show that hypothesis of our results are necessary.

Jordan Ideals and Derivations Satisfying Algebraic Identities

Let $$\mathcal {N}$$ be a 3-prime near-ring with center $$Z(\mathcal {N})$$ and $$\mathcal {J}$$ a nonzero Jordan ideal of $$\mathcal {N}$$ . The aim of this paper is to prove some theorems showing that $$\mathcal {N}$$ must be commutative if it admits a left multiplier F satisfying any one of the following properties: $$(i)\,F(\mathcal {J})\subseteq Z(\mathcal {N})$$ , $$(ii)\,F(\mathcal {J}^{2})\subseteq Z(\mathcal {N})$$ , $$(iii)\,F(ij)+[i, j]\in Z(\mathcal {N})$$ , $$(vi)\,F(ij)-ij+ji\in Z(\mathcal {N})$$ , $$(v)\,F(i\circ j)\in Z(\mathcal {N})$$ and $$(vi)\,F(i)G(j)\in Z(\mathcal {N}),$$ for all $$i, j\in \mathcal {J}.$$ Moreover, we give some examples which show that the hypotheses placed in our results are not superfluous.

Study of Jordan quasigroups and their construction

ABSTRACTJordan quasigroups are commutative quasigroups satisfying the identity . In this paper we discuss the basic properties of Jordan quasigroups and prove that (i) every commutative idempotent quasigroup is Jordan quasigroup, (ii) if a Jordan quasigroup Q is distributive then Q is idempotent, (iii) an idempotent entropic quasigroup satisfies Jordan's identity, (iv) a unipotent quasigroup Q satisfying Jordan's identity satisfies left nuclear square property, (vi) if a quasigroup satisfies LC identity, then alternativity ⇔ Jordan's identity, (vii) for a unipotent Jordan quasigroup Q, and (viii) every Steiner quasigroup is Jordan quasigroup. We also prove some properties of the autotopism of Jordan loops. Moreover, we construct an infinite family of nonassociative Jordan quasigroups whose smallest member is of order 6.

A note on derivations and Jordan ideals of prime rings

Let $F:R\rightarrow R$ be a generalized derivation of a 2-torsion free prime ring $R$ together with a derivation $d.$ In this paper, we show that a nonzero Jordan ideal $J$ of $R$ contains a nonzero ideal of $R$. Further, we use this result to prove that if $F([x, y])\in Z(R)$ for all $x, y\in J, $ then $R$ is commutative. Consequently, it extends a result of Oukhtite, Mamouni and Ashraf.