Quasi-Lie structure of σ-derivations of C[t±1]

Hartwig, Larsson and Silvestrov in [J.T. Hartwig, D. Larsson, S.D. Silvestrov, Deformations of Lie algebras using σ-derivations, J. Algebra 295 (2) (2006) 314–361] defined a bracket on σ-derivations of a commutative algebra. We show that this bracket preserves inner derivations, and based on this obtain structural results providing new insights into σ-derivations on Laurent polynomials in one variable.     

Deformations of Lie algebras using σ-derivations

In this article we develop an approach to deformations of the Witt and Virasoro algebras based on σ-derivations. We show that σ-twisted Jacobi type identity holds for generators of such deformations. For the σ-twisted generalization of Lie algebras modeled by this construction, we develop a theory of central extensions. We show that our approach can be used to construct new deformations of Lie algebras and their central extensions, which in particular include naturally the q-deformations of the Witt and Virasoro algebras associated to q-difference operators, providing also corresponding q-deformed Jacobi identities.     

Invariants of Algebraic Automorphisms

Let R be a prime ring with extended centroid C and let σ be a C-algebraic automorphism of R. We let $R^{(\sigma)}\mathop{=}\limits^{\rm def.}\{x\in R\mid \sigma(x)=x\}$ , the subring of invariants of σ in R, and let Out-deg(σ) and Inn-deg(σ) denote the outer and inner degrees of σ, respectively. In the paper we first prove the nilpotence of the prime radical of R ^(σ) with a bound and characterize the semiprimeness and primeness of R ^(σ). Moreover, we show that if R ^(σ) is a prime PI-ring, then PI-deg(R)?=?PI-deg(R ^(σ)) × Inn-deg(σ).     

Purely Hom-Lie bialgebras

In this paper, we first show that there is a Hom-Lie algebra structure on the set of(σ, σ)-derivations of an associative algebra. Then we construct the dual representation of a representation of a Hom-Lie algebra.We introduce the notions of a Manin triple for Hom-Lie algebras and a purely Hom-Lie bialgebra. Using the coadjoint representation, we show that there is a one-to-one correspondence between Manin triples for Hom-Lie algebras and purely Hom-Lie bialgebras. Finally, we study coboundary purely Hom-Lie bialgebras and construct solutions of the classical Hom-Yang-Baxter equations in some special Hom-Lie algebras using Hom-O-operators.     

Minimal prime ideals of skew polynomial rings and near pseudo-valuation rings

Let R be a ring. We recall that R is called a near pseudo-valuation ring if every minimal prime ideal of R is strongly prime. Let now σ be an automorphism of R and δ a σ-derivation of R. Then R is said to be an almost δ-divided ring if every minimal prime ideal of R is δ-divided. Let R be a Noetherian ring which is also an algebra over ? (? is the field of rational numbers). Let σ be an automorphism of R such that R is a σ(*)-ring and δ a σ-derivation of R such that σ(δ(a)) = δ(σ(a)) for all a ∈ R. Further, if for any strongly prime ideal U of R with σ(U) = U and δ(U) ? δ, U[x; σ, δ] is a strongly prime ideal of R[x; σ, δ], then we prove the following:

1. R is a near pseudo valuation ring if and only if the Ore extension R[x; σ, δ] is a near pseudo valuation ring.
2. R is an almost δ-divided ring if and only if R[x; σ, δ] is an almost δ-divided ring.

     

A generalization of finite dimensionality for modules

Let R be a ring with identity. Let C be a class of R-modules which is closed under submodules and isomorphic images. Define a submodule C of an R-module M to be a C-submodule of M if C ? C. An R-module M is said to be C-finite dimensional if it does not contain an infinite direct sum of non-zero C-submodules of M. Theorem: Let M be a C-finite dimensional R-module. Then there is a uniform bound (the C-dimension of M) on the number of non-zero C-submodules in a direct sum of submodules of M. When C = $\text{M}$_R, we recover the definition of dimension in the sense of Goldie. When C is the class of torsion-free modules relative to a kernel functor σ, we derive the formula: dim M = σ-dim M + dim (σ(M)) where for an R-module N, dim N is the dimension of N in the sense of Goldie and σ-dim N is the dimension of N relative to the class of σ-torsion- free modules. A special case gives a new interpretation of rank of a module as defined by Goldie.     

Cohomology of Hom-Lie superalgebras and q-deformed Witt superalgebra

Hom-Lie algebra (superalgebra) structure appeared naturally in q-deformations, based on σ-derivations of Witt and Virasoro algebras (superalgebras). They are a twisted version of Lie algebras (superalgebras), obtained by deforming the Jacobi identity by a homomorphism. In this paper, we discuss the concept of α ^k -derivation, a representation theory, and provide a cohomology complex of Hom-Lie superalgebras. Moreover, we study central extensions. As application, we compute derivations and the second cohomology group of a twisted osp(1, 2) superalgebra and q-deformed Witt superalgebra.     

Notes on Jordan (σ, τ)*-derivations and Jordan triple (σ, τ)*-derivations

Let R be a 2-torsion free semiprime *-ring, σ, τ two epimorphisms of R and f, d : R → R two additive mappings. In this paper we prove the following results: (i) d is a Jordan (σ, τ)*-derivation if and only if d is a Jordan triple (σ, τ)*-derivation. (ii) f is a generalized Jordan (σ, τ)*-derivation if and only if f is a generalized Jordan triple (σ, τ)*-derivation.     

Orthogonal generalized (σ, τ)-derivations of semiprime rings

This paper abstracts some results of M. Bresar and J. Vukman [1] on the orthogonal derivations of semiprime rings to (σ, τ)-derivations and generalized (σ, τ)-derivations.     

Length problems for automorphisms of modules over local rings

Let R be a local ring and M a free module of a finite rank over R. An element τ ∈ Aut_RM is said to be simple if τ ≠ 1 fixes a hyperplane of M.We shall show that for any σ ∈ Aut_RM there exist a basis X for M and ρ ∈ Aut_RM such that ρ acts as a permutation on X and ρ⁻¹σ is a product of m or less than m simple elements in Aut_RM, where m is the order of the invariant factors of σ modulo the maximal ideal of R.Also we shall investigate the problem treated by E.W. Ellers and H. Ishibashi [Factorizations of transformations over a valuation ring, Linear Algebra Appl. 85 (1987) 17-27], in which they showed that σ is a product of simple elements and gave an upper bound of the smallest number of such factors of σ, whereas in the present paper we will give lower bounds of σ in case that R is a local domain. Moreover we will factorize θσ as a product of symmetries and transvections for some θ the matrix of which is diagonal.     

Generalized derivations as Jordan homomorphisms on lie ideals and right ideals

Let R be a prime ring, L a non-central Lie ideal of R and g a non-zero generalized derivation of R. If g acts as a Jordan homomorphism on L, then either g（x） = x for all x ∈ R, or char（R） = 2, R satisfies the standard identity s4（x1, x2, x3, x4）, L is commutative and u2 ∈ Z（R）, for any u C L. We also examine some consequences of this result related to generalized derivations which act as Jordan homomorphisms on the set [I, I], where I is a non-zero right ideal of R.     

Automorphisms of Ore extensions

Let R be a domain and σ an outer automorphism of R. For any automorphism g of the Ore extension R[t; σ], it is shown that either g(t) = at, g ^?1(t) = bt or g(t) = a, g ^?1(t) = b for some a, b ∈ R. As applications, we show first that R[t; σ] is essentially a quantum plane if R is a commutative domain and if R[t; σ] possesses an automorphism sending t into R. This shows an interesting analogy between the quantum plane and the Weyl algebra. We then determine all ring automorphisms of such R[t; σ].     

Rings and semigroups with permutable zero products

We consider rings R, not necessarily with 1, for which there is a nontrivial permutation σ on n letters such that x₁?x_n=0 implies x_σ(1)?x_σ(n)=0 for all x₁,…,x_n∈R. We prove that this condition alone implies very strong permutability conditions for zero products with sufficiently many factors. To this end we study the infinite sequences of permutation groups P_n(R) consisting of those permutations σ on n letters for which the condition above is satisfied in R. We give the full characterization of such sequences both for rings and for semigroups with 0. This enables us to generalize some recent results by Cohn on reversible rings and by Lambek, Anderson and Camillo on rings and semigroups whose zero products commute. In particular, we prove that rings with permutable zero products satisfy the Köthe conjecture.     

<Emphasis Type="Italic">δ</Emphasis>-derivations of classical Lie superalgebras

We consider the δ-derivations of classical Lie superalgebras and prove that these superalgebras admit nonzero δ-derivations only when δ = 0, ½, 1. The structure of ½-derivations for classical Lie superalgebras is completely determined.     

Rings of continuous functions on σ-frames

This paper deals with the ?-rings RS of all real-valued continuous functions on a completely regular σ-frame. It shows that, in marked contrast with the situation for frames, any ?-ring homomorphism RS→RT results from a σ-frame homomorphism S→T. Further, it proves the analogue of this for integer-valued continuous functions and 0-dimensional σ-frames. In all, this demonstrates that the important classical difference between Alexandroff spaces and Tychonoff spaces with respect to the real-valued continuous functions carries over fully to the pointfree setting - indeed, it adds the integer-valued case which seems to be new in this context.     

A field-theoretic invariant for domains

SoientR ?T des anneaux intègres. D’après Dobbs-Mullins, on pose Λ(T/R) ? sup{λ(k _Q (T)/k _Q∩R (R)) |Q ∈ Spec(T)} où, pour des corpsK?L,λ(L/K) est la longueur maximale d’une chaîne de corps contenus entreK etL. On introduitσ(R):=sup{Λ(T/R)|T est un suranneau deR\. On détermineσ(R) siR′, la clôture intégrale deR, est un anneau de Prüfer et également siR est un anneau de pseudo-valuation. On considère le cas oùσ(R)=1, en particulier siR′ est une extension minimale deR. Plusieurs calculs sont facilités par un résultat sur les carrés cartésiens, et il y a des exemples divers.     

Certain operator algebras induced by 1-derivations in C1-algebras on an indefinite inner product space

We consider operator algebras on an indefinite inner product space, which are induced by ¹-derivations in C¹-algebras, and give some conditions for boundedness of ¹-derivations by using them. Also we show a close relationship between a ¹-derivation and particular subspaces which are invariant under the algebra induced by that ¹-derivation.     

δ-Derivations of simple finite-dimensional Jordan superalgebras

We describe non-trivial δ-derivations of semisimple finite-dimensional Jordan algebras over an algebraically closed field of characteristic not 2, and of simple finite-dimensional Jordan superalgebras over an algebraically closed field of characteristic 0. For these classes of algebras and superalgebras, non-zero δ-derivations are shown to be missing for δ ≠ 0, 1/2, 1, and we give a complete account of 1/2-derivations. Supported by RFBR grant No. 05-01-00230 and by RF Ministry of Education and Science grant No. 11617. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 585–605, September–October, 2007.     

Is Lebesgue measure the only σ-finite invariant Borel measure?

S. Saks and recently R.D. Mauldin asked if every translation invariant σ-finite Borel measure on Rd is a constant multiple of Lebesgue measure. The aim of this paper is to investigate the versions of this question, since surprisingly the answer is “yes and no,” depending on what we mean by Borel measure and by constant. According to a folklore result, if the measure is only defined for Borel sets, then the answer is affirmative. We show that if the measure is defined on a σ-algebra containing the Borel sets, then the answer is negative. However, if we allow the multiplicative constant to be infinity, then the answer is affirmative in this case as well. Moreover, our construction also shows that an isometry invariant σ-finite Borel measure (in the wider sense) on Rd can be non-σ-finite when we restrict it to the Borel sets.     

Symmetry codes and their invariant subcodes

We define and study the invariant subcodes of the symmetry codes in order to be able to determine the algebraic properties of these codes. An infinite family of self-orthogonal rate $\text{1}\text{2}$ codes over GF(3), called symmetry codes, were constructed in [3]. A (2q + 2, q + 1) symmetry code, denoted by C(q), exists whenever q is an odd prime power ≡ ?1, (mod 3). The group of monomial transformations leaving a symmetry code invariant is denoted by G(q). In this paper we construct two subcodes of C(q) denoted by R_σ(q) and R_μ(q). Every vector in R_σ(q) is invariant under a monomial transformation τ in G(q) of odd order s where s divides (q + 1). Also R_μ(q) is invariant under τ but not vector-wise. The dimensions of R_σ(q) and R_μ(q) are determined and relations between these subcodes are given. An isomorphism is constructed between R_σ(q) and a subspace of $\text{W = V}{}_3{}^{\mathrm{<mtext>(2q+2)</mtext><mtext>s</mtext>}}$. It is shown that the image of R_σ(q) is a self-orthogonal subspace of W. The isomorphic images of R_σ(17) (under an order 3 monomial) and R_σ(29) (under an order 5 monomial) are both demonstrated to be equivalent to the (12, 6) Golay code.