Representable Functors and Operations on Rings

Introduction The main aim of this article is to describe the mechanics of certain types of operations on rings (e.g. A-operations on special A-rings or differentiation operators on rings with derivation). En route we meet the very useful notion of a representable functor from rings to rings. If B, R are rings, then the set Hom^(J5,i2) of ring homomorphisms does not, in general, have a ring structure (unlike, for instance, the case where G, H are abelian groups, in which case Homg(G,H) is naturally an abelian group). However, we shall show in § 1 that, if B also has a 'co-ring' structure (in which case we call it a hiring) then this induces a natural ring structure on Homg?(J5,i2). In this case, the functor R \-> Hom^jB, R) is said to be represented by the biring B. In §2, we demonstrate that this functor has a left adjoint, which we denote by R h-» B © R to bring out the analogy with the abelian group case (where H h» G® H is left adjoint to H h> Homg((r,fi)). When, in §4, we come to discuss natural operations on a certain class of rings, there are many constructions we may perform. Given a collection T of operations on a ring, we may give T a ring structure by addition and multiplication of the values of operations. We are also interested in the value of an operation on the sum and product of two elements in terms of its value on those elements. This may be given by imposing a co-ring structure on T in which, for example, the effect of an operation on the sum of two elements is determined by the co-sum' of that operation on the tensor product of the elements. Finally, given two operations on a ring, we may form the composite of one followed by the other. We insist that the identity operation is in T. All these requirements add up to the notion of a 'biring triple' in § 3. In § 4 we discuss the class of rings on which the operations in T act and in §§5, 6 we concentrate on the particular examples afforded by special A-rings and rings with derivation. There follow three appendices which contain material supplementary to that in the main text which would otherwise have interrupted the flow of the exposition.