Composition algebras of polynomials (Q1065124)

A composition algebra A has two operations defined on it, namely, addition and composition (substitution of polynomials). The ring C[x,y,...] of polynomials in the indeterminates x,y,... with coefficients in a commutative ring C is commutative with respect to addition, associative under composition, and one-sided distributive over addition. For two or more indeterminates, composition is not a binary operation. An ideal is defined as the kernel of a homomorphism and a principal ideal composition algebra is one in which for each ideal J there is an element \(r\in J\) such that J is the minimal ideal which contains the element r. The author determines the ideal structure for several special cases, for example C[x] whenever C is the ring of integers, and \(Z_ m[x]\), m odd. The structure of C[x] is given in detail for C a principal ideal ring and such that 2 divides \(c+c^ 2\) for each \(c\in C\). The article is relatively selfcontained with the major results being the terminal member of a sequence of rather brief lemmas.