# Composition ring

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

$\circ :R\times R\rightarrow R$ ## 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.

${\frac {f_{1}}{f_{2}}}\circ g={\frac {f_{1}\circ g}{f_{2}\circ g}}.$ 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 $f_{2}\circ g$ 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.

For a concrete example take the ring ${\mathbb {Z} }[x]$ , considered as the ring of polynomial maps from the integers to itself. A ring endomorphism

$F:{\mathbb {Z} }[x]\rightarrow {\mathbb {Z} }[x]$ $f=F(x)$ $(x^{2}+3x+5)\circ (x-2)=(x-2)^{2}+3(x-2)+5=x^{2}-x+3.$ This example is isomorphic to the given example for R[X] with R equal to $\mathbb {Z}$ , and also to the subring of all functions $\mathbb {Z} \to \mathbb {Z}$ formed by the polynomial functions.