Let φ be a Jordan automorphism of an algebra . The situation when an element a ∈ satisfies $1/2(\phi \left(a\right)+{\phi }^{-1}\left(a\right))=a$ is considered. The result which we obtain implies the Kleinecke-Shirokov theorem and Jacobson’s lemma.

We establish a Künneth formula for some chain complexes in the categories of Fréchet and Banach spaces. We consider a complex of Banach spaces and continuous boundary maps dₙ with closed ranges and prove that Hⁿ(’) ≅ Hₙ()’, where Hₙ()’ is the dual space of the homology group of and Hⁿ(’) is the cohomology group of the dual complex ’. A Künneth formula for chain complexes of nuclear Fréchet spaces and continuous boundary maps with closed ranges is also obtained. This enables us to describe explicitly...

Let $(H,\alpha )$ be a monoidal Hom-Hopf algebra and $(A,\beta )$ a right $(H,\alpha )$-Hom-comodule algebra. We first introduce the notion of a relative Hom-Hopf module and prove that the functor $F$ from the category of relative Hom-Hopf modules to the category of right $(A,\beta )$-Hom-modules has a right adjoint. Furthermore, we prove a Maschke type theorem for the category of relative Hom-Hopf modules. In fact, we give necessary and sufficient conditions for the functor that forgets the $(H,\alpha )$-coaction to be separable. This leads to a generalized...

Let G(X) denote the smallest (von Neumann) regular ring of real-valued functions with domain X that contains C(X), the ring of continuous real-valued functions on a Tikhonov topological space (X,τ). We investigate when G(X) coincides with the ring $C(X,{\tau }_{\delta })$ of continuous real-valued functions on the space $(X,{\tau }_{\delta })$, where ${\tau }_{\delta }$ is the smallest Tikhonov topology on X for which $\tau \subseteq {\tau }_{\delta }$ and $C(X,{\tau }_{\delta })$ is von Neumann regular. The compact and metric spaces for which $G\left(X\right)=C(X,{\tau }_{\delta })$ are characterized. Necessary, and different sufficient, conditions...

We introduce a notion of Morita equivalence for Hilbert C*-modules in terms of the Morita equivalence of the algebras of compact operators on Hilbert C*-modules. We investigate the properties of the new Morita equivalence. We apply our results to study continuous actions of locally compact groups on full Hilbert C*-modules. We also present an extension of Green's theorem in the context of Hilbert C*-modules.

We consider the quotient categories of two categories of modules relative to the Serre classes of modules which are bounded as abelian groups and we prove a Morita type theorem for some equivalences between these quotient categories.

We present a notion of an anti-covariant bialgebra extending the anti-symmetric infinitesimal bialgebra and also provide some equivalent characterizations of it. We also prove that an anti-associative Yang-Baxter pair can produce a special Rota-Baxter system.

If $A{\otimes }_{R,\sigma }V$ and $A{\otimes }_{P,\nu }W$ are two Brzeziński crossed products and Q: W⊗ V → V⊗ W is a linear map satisfying certain properties, we construct a Brzeziński crossed product $A{\otimes }_{S,\theta }\left(V\otimes W\right)$. This construction contains as a particular case the iterated twisted tensor product of algebras.

Leibniz algebras are a non-commutative version of usual Lie algebras. We introduce a notion of (pre)crossed Leibniz algebra which is a simultaneous generalization of notions of representation and two-sided ideal of a Leibniz algebra. We construct the Leibniz algebra of biderivations on crossed Leibniz algebras and we define a non-abelian tensor product of Leibniz algebras. These two notions are adjoint to each other. A (co)homological characterization of these new algebraic objects enables us to...

The weak radical, W-Rad(A) of a non-associative algebra A, has been introduced by A. Rodríguez Palacios in [3] in order to generalize the Johnson's uniqueness of norm theorem to general complete normed non-associative algebras (see also [2] for another application of this notion). In [4], he showed that if A is a semiprime non-associative algebra with DCC on ideals, then W-Rad(A) = 0. In the first part of this paper we give an example of a non-semiprime associative algebra A with DCC on ideals and...

2000 Mathematics Subject Classification: 13N15, 13A50, 16W25.We reduce the Nowicki conjecture on Weitzenböck derivations of polynomial algebras to a well known problem of classical invariant theory.