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

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Focusing on the Wedderburn-Malcev theorem, this work delves into the analysis of radical complements and their properties within commutative and solvable associative algebras. Key topics include representing elements as sums of nilpotent and fully separable elements, alongside the compatibility of the theorem with derived structures. Special attention is given to fully-separable elements and the generalized Jordan decomposition, with practical examples from generalized quaternion algebras and group algebras. The appendix includes proofs related to unitary algebras and solvable unit groups.