In mathematics, a composition ring, introduced in (Adler 1962), is a commutative ring (R, 0, +, −, ·), possibly without an identity 1 (see non-unital ring), together with an operation

such that, for any three elements one has

It is not generally the case that , nor is it generally the case that (or ) has any algebraic relationship to and .


There are a few ways to make a commutative ring R into a composition ring without introducing anything new.

More interesting examples can be formed by defining a composition on another ring constructed from R.

However, as for formal power series, the composition cannot always be defined when the right operand g is a constant: in the formula given the denominator should not be identically zero. One must therefore restrict to a subring of R(X) to have a well-defined composition operation; a suitable subring is given by the rational functions of which the numerator has zero constant term, but the denominator has nonzero constant term. Again this composition ring has no multiplicative unit; if R is a field, it is in fact a subring of the formal power series example.

For a concrete example take the ring , considered as the ring of polynomial maps from the integers to itself. A ring endomorphism

of is determined by the image under of the variable , which we denote by

and this image can be any element of . Therefore, one may consider the elements as endomorphisms and assign , accordingly. One easily verifies that satisfies the above axioms. For example, one has

This example is isomorphic to the given example for R[X] with R equal to , and also to the subring of all functions formed by the polynomial functions.

See also