# Download Cogroups and Co-rings in Categories of Associative Rings by George M. Bergman PDF

By George M. Bergman

This e-book reports representable functors between recognized sorts of algebras. All such functors from associative earrings over a set ring $R$ to every of the kinds of abelian teams, associative earrings, Lie jewelry, and to a number of others are decided. effects also are received on representable functors on different types of teams, semigroups, commutative earrings, and Lie algebras. The e-book contains a ``Symbol index'', which serves as a thesaurus of symbols used and a listing of the pages the place the themes so symbolized are taken care of, and a ``Word and word index''. The authors have strived--and succeeded--in making a quantity that's very effortless.

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**Additional info for Cogroups and Co-rings in Categories of Associative Rings**

**Example text**

1 above, we can let V be the category of partially ordered sets and isotone (<-preserving) maps, and W the variety of distributive lattices with least element 0 and greatest element 1. We find that the structures of partially ordered set and of lattice with least and greatest element on 2 = {0, 1} ''respect" one another, so again we get a pair of right adjoint contravariant functors, POSet(-, 2) and DistLat ' (-, 2), connecting the two categories. , a contravariant equivalence between these subcategories.

Contravariant functors. 8 above) if and only if V is represented by a V-algebra object R of C. Suppose now that we also assume the category C to be a variety of algebras W. „), .... ,xml), .... 9xmn)). If R is such an object, then for any AeOb(W), the set of morphisms W(A, R) is closed under pointwise application of the V-operations of R, and so becomes a V-algebra. , looked at as a V-object of W, is a representing object for the adjoint functor W: V o p —» W. This general situation is examined in [197], [53] and [97].

R for another bifunctor *: C x C - ^ C . Depending on how "product-like" * is, these systems will behave to a greater or lesser degree like algebras. A notable case of this sort (for coalgebras rather than algebras) is the concept which Hopf algebraists call a "coalgebra". These are /c-vector-spaces V with a "comultiplication" A: V-^V^^V. A /c-vector-space having both a structure of associative A:-algebra and a structure of coalgebra in the above sense, satisfying some compatibility conditions, is called a bialgebra over k.