Jordan Ideals Research Articles

Diferential identities on Jordan ideals of rings with involution

Diferential identities on Jordan ideals of rings with involution

Semi-derivations acting on Jordan ideals of rings with involution

Our aim in this paper is to prove some properties of semi-derivations centralizing on Jordan ideals of rings with involution. Some well known results characterizing commutativity of prime rings by derivations have been generalized by using semi-derivations. Moreover, we provide examples to show that our hypotheses are not superfluous.

Commutativity theorems for prime rings with generalized derivations on Jordan ideals

Let R be a 2-torsion free prime ring, J a nonzero Jordan ideal of R and F be a generalized derivation associated with a nonzero derivation d. If F satisfies any one of the following conditions: (i) F(xy)−xy∈Z(R); (ii) F(xy)−yx∈Z(R); (iii) F(x)F(y)−xy∈Z(R); (iv) F(x)F(y)−yx∈Z(R) for all x, y∈J, then R is commutative.

Derivations and Jordan ideals in prime rings

The purpose of this paper is to study derivations satisfying certain differential identities on Jordan ideals of prime rings. Some well known results characterizing commutativity of prime rings by derivations have been generalized by using Jordan ideals. Moreover, we provide examples to show that our results cannot be extended to semi-prime rings.

Generalized derivations on Jordan ideals in prime rings

Let R be a 2-torsion free prime ring with center Z(R), J be a nonzero Jordan ideal also a subring of R, and F be a generalized derivation with associated derivation d. In the present paper, we shall show that J\subseteq Z(R) if any one of the following properties holds: (i) [F(u), u]\in Z(R), (ii) F(u)u = ud(u), (iii) d(u^2)=2F(u)u, (iv) F(u^2)-2uF(u) = d(u^2)-2ud(u), (v) F^2(u)+3d^2(u)=2Fd(u)+2dF(u), (vi) F(u^2) = 2uF(u) for all u \in J.

Generalized derivations centralizing on Jordanideals of rings with involution

A classical result of Posner states that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. In this paper we extend Posner's result to generalized derivations centralizing on Jordan ideals of rings with involution and discuss the related results. Moreover, we provide examples to show that the assumed restriction cannot be relaxed.

Left Multipliers Acting on Jordan Ideals of Rings with Involution

Left Multipliers Acting on Jordan Ideals of Rings with Involution

Semiderivations Satisfying Certain Algebraic Identities on Jordan Ideals

Let be a ring. An additive mapping is called semiderivation of if there exists an endomorphism of such that and , for all in . Here we prove that if is a 2-torsion free -prime ring and a nonzero -Jordan ideal of such that for all , then either is commutative or for all . Moreover, we initiate the study of generalized semiderivations in prime rings.

Commutativity Theorems for *-Prime Rings with Differential Identities on Jordan Ideals

In this paper we explore commutativity of -prime rings in which derivations satisfy certain differential identities on Jordan ideals. Furthermore, examples are given to demonstrate that our results cannot be extended to semiprime rings.

Some Results on (σ,τ)-Left Jordan Ideals in Prime Rings

In this paper we have proved the following results. Let R be a prime ring,U be (σ,τ)-left Joradan ideal of R where σ,τ: R→R be two automorphisms of R and d be a nonzero derivation of R . (1) If (R,a)σ,τ=0, then aZ(R). (2) If aU=0(or Ua=0) and aR , then a=0 or UZ(R). (3) If characteristic of R not equal 2 and U Cσ,τ, then σ(u)+τ(u)Z(R) for all uU.(4) If d(U)=0, dτ=τd and dσ=σd , then σ(u)+τ(u)Z(R) for all uU.

Derivations satisfying certain algebraic identities on Jordan ideals

In this paper, we investigate commutativity of rings with involution in which derivations satisfy certain algebraic identities on Jordan ideals. Moreover, we extend some results for derivations of prime rings to Jordan ideals. Furthermore, an example is given to prove that the ∗-primeness hypothesis is not superfluous.

Jordan Modules and Jordan Ideals of Reflexive Algebras

Let \({\mathcal{L}}\) be a completely distributive subspace lattice on a Banach space and alg \({\mathcal{L}}\) the associated reflexive algebra. Suppose that the following $$\mbox{Condition A:}\dim(F/F\wedge F_-)\ne1\;\; \mbox{for all}\;\;F\in\mathcal{L}$$ holds; note that if \({\mathcal{L}}\) is an atomic Boolean subspace lattice, this condition means that every atom of \({\mathcal{L}}\) has dimension at least two. It is shown that every reflexive Jordan Alg \({\mathcal{L}}\) -module is an associative Alg \({\mathcal{L}}\) -module. We give an example which shows that if the Condition A is removed, then the conclusion is not necessarily true. Moreover, we prove that all reflexive Jordan ideals of Alg \({\mathcal{L}}\) are associative ideals in the case that no the Condition A is assumed. The same conclusions hold for weakly closed Jordan modules and weakly closed Jordan ideals if the rank one subalgebra of Alg \({\mathcal{L}}\) is weakly dense in Alg \({\mathcal{L}}\) .

FOLDING AND UNFOLDING QUANTUM STATES

The reduction of a quantum system ("folding" a quantum system) is described as the reduction of its Lie–Jordan Banach algebra of observables with respect to Lie–Jordan Banach subalgebras and Jordan ideals. The space of states of the reduced Lie–Jordan Banach algebra is described in terms of unreduced states ("unfolding" states) as well as the GNS construction. A few examples are discussed.

Posner’s second theorem for Jordan ideals in rings with involution

Posner’s second theorem states that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. In this paper we extend this result to Jordan ideals of rings with involution. Moreover, some related results are also discussed.

Octonions, E6, and particle physics

In 1934, Jordan et al. gave a necessary algebraic condition, the Jordan identity, for a sensible theory of quantum mechanics. All but one of the algebras that satisfy this condition can be described by Hermitian matrices over the complexes or quaternions. The remaining, exceptional Jordan algebra can be described by 3 × 3 Hermitian matrices over the octonions.We first review properties of the octonions and the exceptional Jordan algebra, including our previous work on the octonionic Jordan eigenvalue problem. We then examine a particular real, noncompact form of the Lie group E6, which preserves determinants in the exceptional Jordan algebra.Finally, we describe a possible symmetry-breaking scenario within E6: first choose one of the octonionic directions to be special, then choose one of the 2×2 submatrices inside the 3×3 matrices to be special. Making only these two choices, we are able to describe many properties of leptons in a natural way. We further speculate on the ways in which quarks might be similarly encoded.

Comparative and International Education: A Bibliography (2008)

Comparative and International Education: A Bibliography (2008)

Armenians’ Dual Identity in Jordan

William Saroyan, a famous American Armenian writer states, “For when two of them meet anywhere in the world, see if they will not create a New Armenia”(video, google.co.uk, 2009). This quote assures that there exist Armenian individuals who are willing to work for the group and its future in a global society. It is this way they have, up to now, succeeded in maintaining their cultural identity, and reproducing it through the years no matter what has happened in their surroundings. For example, the strength of Jordanian-Armenians lies in their solidarity. Through maintaining their unique cultural heritage and the assets that have always belonged to them, the community has, through the years, created an identity that will live on as long as there exist individuals to maintain it. By keeping the group’s infrastructure intact, this dual-identity Armenian and Jordanian in Jordan with different cultural differences, has continued to exist. This identity is reproduced within the group through global education, global socialization techno globalization and a common basis of values that sets it apart from new Diasporas.

On Jordan ideals and matrix rings

For A, a commutative ring, and A = M 2 ( A ) , results by Costa and Keller characterize certain Ep ( 2 , A ) -normalized subgroups of the symplectic group, Sp ( 2 , A ) via structures utilizing Jordan ideals and the notion of radices. The following work creates a Jordan ideal structure theorem for Z 2 -graded rings, A 0 ⊕ A 1 , and a Z 2 -graded matrix algebra. The major theorem is a generalization of Costa and Keller’s previous work on matrix algebras over commutative rings.

Weakly closed Jordan ideals in nest algebras on Banach spaces

We show that weakly closed Jordan ideals in nest algebras on Banach spaces are associative ideals. The decomposability of finite-rank operators in Jordan ideals and the commutants of bimodules are also investigated.

Black September: The 1970–71 Events and their Impact on the Formation of Jordanian National Identity

Soon after the 1967 War, the Palestinian Resistance Movements (PRM) which established themselves in Jordan challenged Jordan's authority presenting a military and even existential threat to Hashemite rule. The prestige of the fedayeen as the primary torchbearers of the struggle against Israel rose dramatically following the Karameh Operation of March 1968, which was perceived as a military victory by the PRM and entrenched its legitimacy. The period between the Karameh raid and September 1970 was characterized by the strengthening of the fedayeen organizations and their entrenchment in Jordan by means of the establishment of autonomous military, political, and social institutions. The bloody events of September 1970–July 1971 induced a change in Jordan's official ideological line and marked a significant step in the coalescence of Jordanian national identity. From then on, the particularist trans-Jordanian identity in Jordan was emphasized more explicitly, along with such other attributes of this identity as Bedouin tribal identity, Islamic identity, and Hashemite identity.