On a certain identity satisfied by a derivation and an arbitrary additive mapping

Summary LetR be a prime ring andd be a nonzero derivation ofR. If an additive mappingf ofR satisfiesd(x)f(x) = 0 for allx R, thenf vanishes on some nonzero left ideal ofR and on some nonzero right ideal ofR.     

On prime ideals and radicals of polynomial rings and graded rings

We extend some known results on radicals and prime ideals from polynomial rings and Laurent polynomial rings to Z$\mathbb{Z}$-graded rings, i.e, rings graded by the additive group of integers. The main of them concerns the Brown–McCoy radical G$G$ and the radical S$S$, which for a given ring A$A$ is defined as the intersection of prime ideals I$I$ of A$A$ such that A/I$A/I$ is a ring with a large center. The studies are related to some open problems on the radicals G$G$ and S$S$ of polynomial rings and situated in the context of Koethe’s problem.     

On generalized Jordan 1-derivation in rings

Let n ⩾ 1 be a fixed integer and let R be an (n + 1)!-torsion free 1-ring with identity element e. If F, d:R → R are two additive mappings satisfying F(xⁿ⁺¹) = F(x)(x¹)ⁿ + xd(x)(x¹)ⁿ⁻¹ + x²d(x)(x¹)ⁿ⁻²+ ⋯ +xⁿd(x) for all x ∈ R, then d is a Jordan 1-derivation and F is a generalized Jordan 1-derivation on R.     

On the annihilator of commutators with derivation in prime rings

SianoR un anello primo di caratteristica differente da 2,d una derivazione non nulla diR, L un ideale di Lie non centrale diR, a ∈ R. Sea[d(u), u] = 0, per ogni scelta diu ∈ L, alloraa = 0.     

On a theorem of J. Vukman

Summary 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. Theorem.Let R be a non-commutative prime ring of characteristic different from two, and I a nonzero two-sided ideal of R. Let D: R × R R be a symmetric bi-derivation. If [D(x, x), x] lies in the center of R for all x I, then D = 0.     

On generalized Lie derivations of Lie ideals of prime algebras

Let A be a prime algebra of characteristic not 2 with extended centroid C, let R be a noncentral Lie ideal of A and let B be the subalgebra of A generated by R. If f,d:R→A are linear maps satisfying that

     

On functional identities and d-free subsets of rings. II

We show that a map in several variables on a prime ring satisfying an identity of polynomial type must be a quasi-polynomial (i.e., a polynomial in noncommutative variables whose coefficients are Martindale centroid valued functions)provided that the ring does not satisfy a standard identity of low degree. Obtained results have applications to the study of Lie maps of prime rings (Lie ideals of prime rings and skew elements of prime rings with involution)and to the study of Lie-admissible algebras and Lie homomorphisms of Lie algebras of Poisson algebras.     

On some equations in prime rings

The main purpose of this paper is to prove the following result. Let R be a prime ring of characteristic different from two and let T : R → R be an additive mapping satisfying the relation T(x ³) = T(x)x ² − xT(x)x + x ² T(x) for all x ∈ R. In this case T is of the form 4T(x) = qx + xq, where q is some fixed element from the symmetric Martindale ring of quotients. This result makes it possible to solve some functional equations in prime rings with involution which are related to bicircular projections.     

Differential commutator identities

I.N. Herstein proved that if R is a prime ring satisfying a differential identity , with d a nonzero derivation of R, then R embeds isomorphically in M₂(F) for F a field. We consider a natural generalization of this result for the class of polynomials E_n(X)=[E_n-1(x₁,…,x_n-1),x_n]. Using matrix computations, we prove that if R satisfies a differential identity , or with some restrictions, then R must embed in M₂(F), but that differential identities using [[E_n,E_m],E_s] with m,n,s>1 need not force R to embed in M₂(F). These results hold if the expressions are identities for a noncommutative Lie ideal of R, rather than for R itself.     

FINITE-DIMENSIONAL NON-COMMUTATIVE POISSON ALGEBRAS. II

Here is a structure theorem of a finite-dimensional non-commutative Poisson algebra A. A nice element ε of A will be found, so that the Lie module action of an element of a large Poisson subalgebra of A on A is described in terms of ε and the ordinary associative commutator. Consequently, we can figure out a structure of A when the Jacobson radical rad A satisfies (rad A)² = 0. This structure theorem leads us to a classification of the finite-dimensional simple Poisson A-modules.     

Prime rings with semigroup generalized identity

We show that a prime ring satisfies a nontrivial semigroup generalized identity if and only if its central closure is a primitive ring with nonzero socle and the associated skew field is a field.     

On the Lie algebra of skew-symmetric elements of an enveloping algebra

Let L be a restricted Lie algebra over a field of characteristic p>2 and denote by u(L) its restricted enveloping algebra. We establish when the Lie algebra of skew-symmetric elements of u(L) under the principal involution is solvable, nilpotent, or satisfies an Engel condition.     

On bimeasurings II

We answer some questions posed in [L. Grunenfelder, M. Mastnak, On bimeasurings, J. Pure Appl. Algebra 204 (2006) 258-269] concerning the universal bimeasuring bialgebra via a construction of suitable subcoalgebras of the universal measuring coalgebra.     

On Jordan Triple Derivations and Related Mappings

In this article we study certain functional equations and systems of functional equations related to (generalized) derivations on semiprime rings. In particular, we prove that any generalized Jordan triple derivation on a 2- torsion free semiprime ring is a generalized derivation. We also prove that any (generalized) Jordan triple *-derivation on a 2-torsion free semiprime *-ring is a (generalized) Jordan *-derivation. The second author was supported in part by the Ministry of Science, Education and Sports of the Republic of Croatia (project No. 037-0372784-2757).     

Connected quantized Weyl algebras and quantum cluster algebras

For an algebraically closed field $\mathbb{K}$, we investigate a class of noncommutative $\mathbb{K}$-algebras called connected quantized Weyl algebras. Such an algebra has a PBW basis for a set of generators $\{x_1,\dots ,x_n\}$ such that each pair satisfies a relation of the form $x_i{x}_j=q_{ij}{x}_j{x}_i+r_{ij}$, where $q_{ij}\in {\mathbb{K}}^{?}$ and $r_{ij}\in \mathbb{K}$, with, in some sense, sufficiently many pairs for which $r_{ij}\ne 0$. For such an algebra it turns out that there is a single parameter q such that each $q_{ij}=q^{\pm 1}$. Assuming that $q\ne \pm 1$, we classify connected quantized Weyl algebras, showing that there are two types linear and cyclic. When q is not a root of unity we determine the prime spectra for each type. The linear case is the easier, although the result depends on the parity of n, and all prime ideals are completely prime. In the cyclic case, which can only occur if n is odd, there are prime ideals for which the factors have arbitrarily large Goldie rank.We apply connected quantized Weyl algebras to obtain presentations of two classes of quantum cluster algebras. Let $n\ge 3$ be an odd integer. We present the quantum cluster algebra of a Dynkin quiver of type $A_{n?1}$ as a factor of a linear connected quantized Weyl algebra by an ideal generated by a central element. We also consider the quiver $P_{n+1}^{(1)}$ identified by Fordy and Marsh in their analysis of periodic quiver mutation. When n is odd, we show that the quantum cluster algebra of this quiver is generated by a cyclic connected quantized Weyl algebra in n variables and one further generator. We also present it as the factor of an iterated skew polynomial algebra in $n+2$ variables by an ideal generated by a central element. For both classes, the quantum cluster algebras are simple noetherian.We establish Poisson analogues of the results on prime ideals and quantum cluster algebras. We determine the Poisson prime spectra for the semiclassical limits of the linear and cyclic connected quantized Weyl algebras and show that, when n is odd, the cluster algebras of $A_{n?1}$ and $P_{n+1}^{(1)}$ are simple Poisson algebras that can each be presented as a Poisson factor of a polynomial algebra, with an appropriate Poisson bracket, by a principal ideal generated by a Poisson central element.     

PRIME IDEALS IN RINGS WITH INVOLUTION

Abstract

Let R be a ring with involution *. We show that if R is a *-prime ring which is not a prime ring, then R is “essentially” a direct product of two prime rings. Moreover, if P is a *-prime *-ideal of R, which is not a prime ideal of R, and X is minimal among prime ideals of R containing P, then P is a prime ideal of X, P = X ∩ X* and either: (1) P is essential in X and X is essential in R; or (2) for any relative complement C of P in X, then C* is a relative complement of X in R. Further characterizations of *-primeness are provided.     

On additive polynomials and certain maximal curves

We show that a maximal curve over F_q² given by an equation A(X)=F(Y), where A(X)∈F_q²[X] is additive and separable and where F(Y)∈F_q²[Y] has degree m prime to the characteristic p, is such that all roots of A(X) belong to F_q². In the particular case where F(Y)=Y^m, we show that the degree m is a divisor of q+1.