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 one has It is not generally the case that , nor is it generally the case that (or ) has any algebraic relationship to and . Examples There are a few ways to make a commutative ring R into a composition ring without introducing anything new. Composition may be defined by for all f,g. The resulting composition ring is a rather uninteresting. Composition may be defined by for all f,g. This is the composition rule for constant functions. If R is a boolean ring, then multiplication may double as composition: for all f,g. More interesting examples can be formed by defining a composition on another ring constructed from R. The polynomial ring R[X] is a composition ring where for all . The formal power series ring R also has a substitution operation, but it is only defined if the series g being substituted has zero constant term (if not, the constant term of the result would be given by an infinite series with arbitrary coefficients). Therefore, the subset of R formed by power series with zero constant coefficient can be made into a composition ring with composition given by the same substitution rule as for polynomials. Since nonzero constant series are absent, this composition ring does not have a multiplicative unit. If R is an integral domain, the field R(X) of rational functions also has a substitution operation derived from that of polynomials: substituting a fraction g1/g2 for X into a polynomial of degree n gives a rational function with denominator , and substituting into a fraction is given by 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. The set of all functions from R to R under