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

By George M. Bergman

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

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A nonunital k-ring will mean a k-bimodule R given with a nonunital associative ring structure such that the additive group structures as ring and k-bimodule are the same, and the ring and bimodule multiplications are related by the identities (ax)y = oc(xy), (xa)y - x(ay), (xy)a = x(ya), (x,yeR, aek). A homomorphism of nonunital k-rings will mean a map of underlying sets which is a homomorphism both of rings and of k-bimodules. The variety of nonunital rings will be denoted Ring, and that of nonunital k-rings will be denoted &-Ring.

14 (iii)=>(i), to show W has a left adjoint it suffices to show that there exist a set X, and a set of relations Y in indeterminates from X, such that for any 5 in V, we can describe the functor I W(—)l as taking each S to the set of X-tuples of members of S satisfying the relations Y. But the statement that W respects underlying sets means that IW(5)I can, in fact, be described as the set of all 1-tuples of members of S satisfying the empty set of relations. • This Corollary is applicable to such constructions as (i) the underlying-set functor Uy\ V -» Set for any variety V, the left adjoint of which, as we have already noted, is the free algebra construction, (ii) the "commutator brackets" functor from associative /c-algebras to Lie algebras over k, whose left adjoint is the universal enveloping algebra construction, (iii) the functor taking associative /c-algebras A to their underlying /c-vector-spaces, whose left adjoint is the tensor algebra construction, and (iv) the functor taking associative /:-algebras to their underlying multiplicative semigroups, whose left adjoint is the semigroup-algebra construction (cf.

10. Let C be a category with finite coproducts, and Y a variety of algebras, consisting of all algebras of a specified type which satisfy a specified system of identities Z. Then a co-V object of C will mean a coalgebra R in C of the type of Y, satisfying the following equivalent conditions: (i) R satisfies the {dualized) diagrammatic conditions ("coidentities") ponding to the identities of E. corres- (ii) For each identity u - v in X, say in n variables, if we form the n-fold coproduct \R\ i i ...

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