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Provides a comprehensive introduction to module theory and the related part of ring theory, including original results as well as more recent work. The text starts from a basic understanding of linear algebra, the theory is presented, accompanied by proofs.

This presentation covers known and new results on extending modules and related notions. An extending module is a module for which every submodule is essential in a direct summand. Semisimple modules, uniform modules and injective modules are examples of extending modules.

After describing the module-theoretic aspects of coalgebras over commutative rings, this volume defines corings as coalgebras for non-commutative rings. Topics covered include module-theoretic aspects of corings (such as the relation of comodules to special subcategories of modules: sigma-type categories); connections between corings and extensions of rings; properties of new examples of corings associated to entwining structures; generalizations of bialgebras such as bialgebroids and weak bialgebras; and the appearance of corings in non-commutative geometry.