In mathematics, a composition ring, introduced in , is a commutative ring (R, 0, +, , ·), possibly without an identity 1, together with an operation

one has

It is not, nor.

Examples

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

• for all f,g.

• for all f,g. This is the composition rule for constant functions.

• If R is a boolean ring 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.

• 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, 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 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 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.

, considered as the ring of polynomial maps from the integers to itself. A ring endomorphism

, which we denote by

satisfies the above axioms. For example, one has

This example is isomorphic to the given example for R[X] with R formed by the polynomial functions.

See also

• Composition operator

• Polynomial decomposition

• Jabotinsky matrix

References