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.

Preface; Notations; 1. Coalgebras and comodules; 2. Bialgebras and hopf algebras; 3. Corings and comodules; 4. Corings and extensions of rings; 5. Corings and entwining structures; 6. Weak corings and entwining; Appendix.