%0 Book
%A Sven Bodo Wirsing
%D 2018
%C Hamburg, Germany
%I Anchor Academic Publishing
%@ 9783960677215
%T Separability within commutative and solvable associative algebras. Under consideration of non-unitary algebras. With 401 exercises
%U https://m.anchor-publishing.com/document/452479
%X Within the context of the Wedderburn-Malcev theorem a radical complement exists and all complements are conjugated. The main topics of this work are to analyze the Determination of a (all) radical complements, the representation of an element as the sum of a nilpotent and fully separable element and the compatibility of the Wedderburn-Malcev theorem with derived structures. Answers are presented in details for commutative and solvable associative algebras. Within the analysis the set of fully-separable elements and the generalized Jordan decomposition are of special interest. We provide examples based on generalized quaternion algebras, group algebras and algebras of traingular matrices over a field. The results (and also the theorem of Wedderburn-Malcev and Taft) are transferred to non-unitary algebras by using the star-composition and the adjunction of an unit. Within the App endix we present proofs for the Wedderburn-Malcev theorem for unitary algebras, for Taft's theorem on G-invariant radical complements for unitary algebras and for a theorem of Bauer concerning solvable unit groups of associative algebras.
%K associative, algebra, commutative, solvable, non-unitary, Wedderburn-Malcev theorem
%G English