The notion of coring is a generalization of a -coalgebra. While for a coalgebra must be a commutative ring (often a field), a coring is defined over a general noncommutative ring or even an associative algebra .
Whereas a coalgebra structure is defined on a -module (if is a field, it is a vector space) – which may be regarded as a central -bimodule – a coring structure is defined on a general bimodule over a general ring.
More generally, fix a ground commutative ring . Corings will be now over -algebras. So a coring will mean a pair where is an -algebra and an -coring.
Let be a morphism of rings and an -coring. Then the --bimodule has an induced structure of a -coring with comultiplication
and the counit
A morphism is a pair where
The last two conditions can be said that the base ring extension coring of maps to (via map induced by ) as a morphism of -corings.
The classical example of a coring is the Sweedler coring corresponding to an extension of unital rings. The category of descent data for this extension is equivalent to the category of comodules over the Sweedler coring.
Corings are in general useful for the treatment of descent in noncommutative algebraic geometry.
Another major class of examples are the so-called matrix coring?s.
The notion of an -coring is introduced by M. Sweedler and recently lived through a renaissance in works of T. Brzeziński, R. Wisbauer, G. Böhm, L. Kaoutit, Gómez-Torrecillas, S. Caenepeel, J. Y. Abuhlail, J. Vercruysse and others, including the creation of Galois theory for corings. Some prefer to speak about -cocategories.
There is already a monograph:
L. El Kaoutit, J. Gomez-Torrecillas, On the set of grouplikes of a coring, arxiv/0901.4291
T. Brzeziński, Flat connections and (co)modules, [in:] New Techniques in Hopf Algebras and Graded Ring Theory, S Caenepeel and F Van Oystaeyen (eds), Universa Press, Wetteren, 2007 pp. 35-52 arxiv:math.QA/0608170
Lars Kadison, Depth two and Galois coring, math.RA/0408155
George M. Bergman, Adam O. Hausknecht, Cogroups and co-rings in categories of associative rings, A.M.S. Math. Surveys and Monographs 45, ix+388 pp., 1996; ISBN 0-8218-0495-2. MR 97k:16001 errata and updates.
T. Brzeziński, L. Kadison, R. Wisbauer, On coseparable and biseparable corings, Hopf algebras in noncommutative geometry and physics, 71–87, Lecture Notes in Pure and Appl. Math., 239, Dekker, New York, 2005.
There is a generalization of corings: