We study a class of strongly solvable modes, called differential modes. We characterize abelian algebras in this class and prove that all of them are quasi-affine, i.e., they are subreducts of modules over commutative rings.

Let $ℛ$ be a commutative ring, $𝒢$ be a generalized matrix algebra over $ℛ$ with weakly loyal bimodule and $𝒵\left(𝒢\right)$ be the center of $𝒢$. Suppose that $𝔮:𝒢\times 𝒢\to 𝒢$ is an $ℛ$-bilinear mapping and that ${𝔗}_{𝔮}:𝒢\to 𝒢$ is a trace of $𝔮$. The aim of this article is to describe the form of ${𝔗}_{𝔮}$ satisfying the centralizing condition $[{𝔗}_{𝔮}\left(x\right),x]\in 𝒵\left(𝒢\right)$ (and commuting condition $[{𝔗}_{𝔮}\left(x\right),x]=0$) for all $x\in 𝒢$. More precisely, we will revisit the question of when the centralizing trace (and commuting trace) ${𝔗}_{𝔮}$ has the so-called proper form from a new perspective. Using the aforementioned...

Let $𝔄=\left(\begin{array}{cc}𝒜& ℳ\\ & ℬ\end{array}\right)$ be the triangular algebra consisting of unital algebras $𝒜$ and $ℬ$ over a commutative ring $R$ with identity $1$ and $ℳ$ be a unital $(𝒜,ℬ)$-bimodule. An additive subgroup $𝔏$ of $𝔄$ is said to be a Lie ideal of $𝔄$ if $[𝔏,𝔄]\subseteq 𝔏$. A non-central square closed Lie ideal $𝔏$ of $𝔄$ is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on $𝔄$, every generalized Jordan triple higher derivation of $𝔏$ into $𝔄$ is a generalized higher derivation of $𝔏$ into $𝔄$.

We give a sufficient condition under which any Jordan automorphism of a triangular algebra is either an automorphism or an anti-automorphism.

In this paper we prove that every bijection preserving Lie products from a triangular algebra onto a normal triangular algebra is additive modulo centre. As an application, we described the form of bijections preserving Lie products on nest algebras and block upper triangular matrix algebras.

2000 Mathematics Subject Classification: 15A69, 15A78.In [3] we present the construction of the semi-symmetric algebra [χ](E) of a module E over a commutative ring K with unit, which generalizes the tensor algebra T(E), the symmetric algebra S(E), and the exterior algebra ∧(E), deduce some of its functorial properties, and prove a classification theorem. In the present paper we continue the study of the semi-symmetric algebra and discuss its graded dual, the corresponding canonical bilinear form,...