# Separability within commutative and solvable associative algebras. Under consideration of non-unitary algebras. With 401 exercises

Research Paper (postgraduate) 2018 0 Pages

## Summary

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.

## Details

- Pages
- 0
- Type of Edition
- Erstauflage
- Year
- 2018
- ISBN (eBook)
- 9783960677215
- ISBN (Book)
- 9783960672210
- Language
- English
- Catalog Number
- v452479
- Grade
- Tags
- associative algebra commutative solvable non-unitary Wedderburn-Malcev theorem