Polynomial and transformation composition rings (Q1580083)

Composition rings are sets \(C\) with three operations: \(+\), \(\cdot\) and \(\circ\), such that \((C,+,\cdot)\) is a ring, \((C,+,\circ)\) is a near-ring and \((ab)\circ c=(a\circ c)\cdot(b\circ c)\). A typical case is the set of all polynomials or power series over a ring, with the usual operations of addition and multiplication, and \(\circ\) as composition of functions. Another example is \(M(R)\), the near-ring of all maps from a ring \(R\) to itself, with pointwise addition and multiplication, and composition of functions. These two examples are generalized to give the concepts of polynomial and transformation composition near-rings. The base \(B\) of an arbitrary composition ring is defined as the set of elements \(v\in C\) distributive over \((C,+)\) and such that \(v\circ(ab)=(v\circ a)b\). For any composition ring, its base \(B\) is a subring of the zero-symmetric part of the near-ring and has many interesting properties. A good deal of information about the situation is provided and a wide range of examples are detailed. Composition rings of polynomial type are then examined, and again a range of examples are given. The next concept to be introduced is that of a semi-constant. A number of results concerning this idea are proved. This is then linked up to the definition of composition rings of transformation type. Again results and examples are given. The last section looks at ideals in composition rings of both types. There are many results here, many of them on maximal ideals. This paper provides a lot of information on the structure of general composition rings.