Separability Within Commutative And Solvable Associative Algebras Under Consideration Of Non Unitary Algebras With 401 Exercises

Download or read book Separability within commutative and solvable associative algebras. Under consideration of non-unitary algebras. With 401 exercises written by Sven Bodo Wirsing and published by Anchor Academic Publishing. This book was released on 2018-12-12 with total page 257 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.