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 ...