Briefly, a composition
algebra A involves two operations: addition and composition (substitution of
polynomials). Let C be an arbitrary commutative ring, and C[x,y,…] the ring of
polynomials in the indeterminates x,y,… with coefficients from C. Addition of
polynomials is commutative; composition is associative, and is distributive (on one
side) over addition. (Notice that if the number of indeterminates is greater
than 1, the operation of composition is not a binary operation.) We find
the ideal structure of A in some special cases. In particular, the ideals of A
are all principal (generated by a single element) if C is a principal ideal
ring (e.g. Z) and the number of variables is 1 : A = (C[x],+,∘), provided
further that for all c ∈ C,2|c + c2. [An example is the algebraic integers in
Q().]
