Aspects of the circle composition operation in rings

In this thesis we will investigate some of the properties of the circle composition (or adjoint) operation in rings, where the operation o is defined by aob = a + b + ab. In arbitrary rings, R, the properties of addition and multiplication imply that (R, o) is a semigroup; in certain classes of rings this semigroup has additional properties and we shall examine a few of these. Our main concern will be commutative quasiregular (Jacobson radical) rings. In such rings (R, o) is an abelian group, giving R a second such structure besides (R,+). It seems a natural question to ask if these group structures can ever be isomorphic. The zero rings, in which multiplication is trivial, obviously have this property since the additive and circle composition groups coincide; thus the class, K, of rings having isomorphic additive and circle composition groups is non-empty. There are also non-trivial examples and we illustrate the construction of some, including the so-called quasifields which are constructed on partially ordered sets, and examples which use finite groups for addition. It might be suspected that for these less trivial examples the isomorphism between addition and circle composition will still force multiplication to behave in a nearly trivial way, so that perhaps such rings are nil or nilpotent. This need not be the case as there is a ring in K which has no zero divisors. In fact, we show that there exist rings in K which are nilpotent but not zero rings, nil but not nilpotent, and quasiregular without being nil. We will also consider the algebraic properties of the class K, including the question of its inheritance under ring theoretic constructions. In particular, we show that K is not a radical class, that it is closed under direct products, but that it is not hereditary and that it is not closed under homomorphisms nor taking quasiregular subrings. There are, however, certain subclasses of K which are better behaved, including, for example, rings which are algebras over Zp or Q and the rings constructed on certain finite groups. For commutative nilpotent rings we prove the existence of a polynomial homomorphism between the additive and circle composition groups, which in certain circumstances will be an isomorphism. We show, too, that all finitely generated nilpotent Q-algebras and Z-algebras are in K. The former result allows us to demonstrate that all commutative nil Q-algebras are in K. We conclude by considering a family of ring examples in which the circle composition semigroup is regular. Our construction is developed from the idea behind the quasifield construction and also generalised power series rings. We investigate the existence of nilpotence in such rings, and show that, like K, the class of rings in which (R,o) is a regular semigroup is not a radical class. This result also holds for the stronger property that (R,o) is a union of groups.