pointwise addition and multiplication, and with given by composition of functions, is a composition ring. There are numerous variations of this idea, such as the ring of continuous, smooth, holomorphic, or polynomial functions from a ring to itself, when these concepts makes sense. 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 Composition operator Polynomial decomposition Carleman matrix References Algebraic structures Ring theory