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prefacechapter 1  groups  1.1  semigroups, monoids and groups  1.2  subgroups  1.3  the action of a group on a set  1.4  the sylow theorem  1.5  homomorphisms and normal subgroups  1.6  direct products and direct sums  1.7  simple groups  1.8  nilpotent groups and solvable groupschapter 2  modules  2.1  rings and ring homomorphisms  2.2  modules and free modules  2.3  projective modules and injective modules  2.4  homological dimensions and semisimple rings  2.5  tensor product and weak dimension  2.6  localization  2.7  noetherian modules and ufd  2.8  finitely generated modules over a pidchapter 3  fields and galois theory of equations  3.1  extensions of fields  3.2  splitting fields, and normality  3.3  the main theorem of galois theory  3.4  radical extensions  3.5  construction with straight-edge and compass  3.6  the hilbert nullstellensatzchapter 4  introduction of various algebras  4.1  associative algebras  4.2  coassociative coalgebras and hopf algebras  4.3  nonassociative algebraschapter 5  category  5.1  category: direct limits and colimits  5.2  functors and natural transformations  5.3  abelian categories and homological groupsbibliographyindex