In this paper, we characterize the units of skew PBW extensions over compatible rings. With this aim, we recall the transfer of the property of being 2-primal for these extensions. As a consequence of our treatment, the results established here generalize those corresponding for commutative rings and Ore extensions of injective type. In this way, we present new results for several noncommutative rings of polynomial type not considered before in the literature.

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We study nilpotent Lie algebras endowed with a complex structure and a quadratic structure which is pseudo-Hermitian for the given complex structure. We propose several methods to construct such Lie algebras and describe a method of double extension by planes to get an inductive description of all of them. As an application, we give a complete classification of nilpotent quadratic Lie algebras where the metric is Lorentz-Hermitian and we fully classify all nilpotent pseudo-Hermitian quadratic Lie algebras up to dimension 8 and their inequivalent pseudo-Hermitian metrics.

Let M R be a module and σ an endomorphism of R. Let m ∈ M and a ∈ R, we say that M R satisfies the condition C 1 (respectively, C 2 ), if ma = 0 implies mσ(a) = 0 (respectively, mσ(a) = 0 implies ma = 0). We show that if M R is p.q.-Baer then so is M [x; σ] R[x;σ] whenever M R satisfies the condition C 2 , and the converse holds when M R satisfies the condition C 1 . Also, if M R satisfies C 2 and σ-skew Armendariz, then M R is a p.p.-module if and only) is a p.p.-module. Many generalizations are obtained and more results are found when M R is a semicommutative module.