Jordan τ-derivations of Prime Rings

Let R be a prime ring which is not commutative, with maximal symmetric ring of quotients Q _ms (R), and let τ be an anti-automorphism of R. An additive map δ: R → Q _ms (R) is called a Jordan τ-derivation if δ(x ²) = δ(x)x ^τ + xδ(x) for all x ∈ R. A Jordan τ-derivation of R is called X-inner if it is of the form x → ax ^τ ? xa for x ∈ R, where a ∈ Q _ms (R). It is proved that any Jordan τ-derivation of R is X-inner if either R is not a GPI-ring or R is a PI-ring except when charR = 2 and dim  _C RC = 4, where C is the extended centroid of R.     

On triangular algebras with noncommutative diagonals

We construct a triangular algebra whose diagonals form a noncommutative algebra and its lattice of invariant projections contains only two nontrivial projections. Moreover we prove that our triangular algebra is maximal.     

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.     

On σ-commutativity in Menger algebras of n-place functions

We present an abstract characterization of various types of Menger algebras of n-place functions allowing certain permutations of variables.     

σ-Contractible and σ-biprojective Banach algebras

Abstract

The notion of σ-amenability for Banach algebras and its related notions were introduced and extensively studied by M.S. Moslehian and A.N. Motlagh in [10]. We develop these notions parallel to the amenability of Banach algebras introduced by B.E. Johnson in [5]. Briefly, we investigate σ-contractibility and σ-biprojectivity of Banach algebras, which are extensions of usual notions of contractibility and biprojectivity, respectively, where σ is a bounded endomorphism of the corresponding Banach algebra. We also give the notion σ-diagonal. Then we verify relations between σ-contractibility, σ-biprojectivity and the existence of a σ-diagonal for a Banach algebra, when σ has dense range or is an idempotent. Moreover, we obtain some hereditary properties of these concepts.     

Isotopes of prime (−1, 1)-and Jordan algebras

We deal with adjoint commutator and Jordan algebras of isotopes of prime strictly (−1, 1)-algebras. It is proved that a system of identities of the form [x ₁, x ₂, x ₂, x ₃,…, x _n ] for n = 2,.…, 5 is discernible on isotopes of prime (−1, 1)-algebras. Also it is shown that adjoint Jordan algebras for suitable isotopes of prime (−1, 1)-algebras may possess distinct sets of identities. In particular, isotopes of a prime Jordan monster have different sets of identities in general.     

Criterion for a chain of σ-processes to be the composition of triangular chains

The purpose of this paper is to establish a topological criterion for a chain of spheres in a two-dimensional complex manifold to be a composition of so-called triangular chains. Explicit formulas are given for the relevant topological characteristics of chains. Certain corollaries of this criterion related to the Jacobian conjecture are discussed. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 643–653, May, 1999.     

Lattice orders on 2 × 2 triangular matrix algebras

We construct all the lattice orders on a 2 × 2 triangular matrix algebra over a totally ordered field that make it into a lattice-ordered algebra. It is shown that every lattice order in which the identity matrix is not positive may be obtained from a lattice order in which the identity matrix is positive.     

Approximations of ternary Jordan homomorphisms and derivations in multi-C ? ternary algebras

Using fixed point methods, we prove the generalized Hyers–Ulam stability of homomorphisms in multi-C ^? ternary algebras and of derivations on multi-C ^? ternary algebras for the additive functional equation $$\sum_{i=1}^{m}f \bigg(mx_i+\sum_{j=1,\ j\ne i}^{m}x_j\bigg)+ f\bigg(\sum_{i=1}^{m}x_i\bigg)= 2f\bigg(\sum_{i=1}^{m}mx_i\bigg) \quad (m\in {\mathbb{N}},\ m\geqq2).$$      

Jordan–Hölder theorems for derived module categories of piecewise hereditary algebras

A Jordan–Hölder theorem is established for derived module categories of piecewise hereditary algebras, in particular for representations of quivers and for hereditary abelian categories of a geometric nature. The resulting composition series of derived categories are shown to be independent of the choice of bounded or unbounded derived module categories, and also of the choice of finitely generated or arbitrary modules.     

The classification of N-dimensional non-associative Jordan algebras with (N − 3)-dimensional annihilator

The paper is devoted to give a complete classification of all n-dimensional non-associative Jordan algebras with (n?3)-dimensional annihilator over an algebraically closed field of characteristic ≠2. We also give a complete classification of all n-dimensional Jordan algebras with (n?1)- and (n?2)-dimensional annihilator.     

A basic functional identity with applications to Jordan σ-biderivations

Let R be a noncommutative prime ring with extended centroid C and maximal left ring of quotients Q_ml(R). The aim of the paper is to study a basic functional identity concerning bi-additive maps on R. Precisely, it is proved that a bi-additive map B:R×R→Q_ml(R) satisfying [B(x,y),[x,y]] = 0 for all x,y∈R must be of the form (x,y)?λ[x,y]+μ(x,y) for x,y∈R, where λ∈C and μ:R×R→C is a bi-additive map. As applications to the theorem, Jordan σ-biderivations with σ an epimorphism and additive commuting maps on noncommutative Lie ideals of R are characterized.     

Cohen–Macaulay Auslander algebras of skewed-gentle algebras

For any skewed-gentle algebra, we characterize its indecomposable Gorenstein projective modules explicitly and describe its Cohen–Macaulay Auslander algebra. We prove that skewed-gentle algebras are always Gorenstein, which is independent of the characteristic of the ground field, and the Cohen–Macaulay Auslander algebras of skewed-gentle algebras are also skewed-gentle algebras.