Composition ring

In mathematics, a composition ring, introduced in , 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 formula_2 one has
It is "not" generally the case that formula_6, "nor" is it generally the case that formula_7 (or formula_8) has any algebraic relationship to formula_9 and formula_10.
Examples.
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".
For a concrete example take the ring formula_20, considered as the ring of polynomial maps from the integers to itself. A ring endomorphism
of formula_20 is determined by the image under formula_23 of the variable formula_24, which we denote by
and this image formula_26 can be any element of formula_20. Therefore, one may consider the elements formula_28 as endomorphisms and assign formula_29, accordingly. One easily verifies that formula_20 satisfies the above axioms. For example, one has
This example is isomorphic to the given example for "R"["X"] with "R" equal to formula_32, and also to the subring of all functions formula_33 formed by the polynomial functions.