Lehrbuch der Algebra 1

The book covers fundamental concepts in set theory, including sets, mappings, and cardinal numbers, as well as the power of infinite sets. It delves into Zorn's Lemma and explores groups and rings, discussing operations, subgroups, cyclic groups, and special ring elements. The text also examines modules and algebras, detailing modules, submodules, ideals, and linear equations, alongside concepts of linear independence and vector space dimensions. Homomorphisms of groups and rings are addressed, highlighting isomorphisms, residue classes, and the operation of monoids. The book further investigates Sylow's theorems, quadratic residues, and free groups. In the realm of modules, it discusses homomorphisms, basic theorems, and direct sums, as well as matrices and dualization. Determinants are thoroughly examined, including permutations, multilinear mappings, and determinants of endomorphisms and square matrices. The text also covers various determinant properties and norms in algebras. Additionally, it includes a literature section, a list of symbols, and an index of names and subjects, providing a comprehensive resource for understanding these mathematical concepts.