nLab composition ring

This entry is about rings which mimic the composition operation on endo-functions on the ground ring. For the unrelated notion of algebras mimicking the composition of sums of squares see at composition algebra.

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Idea

The notion of composition rings is an abstraction of the structure present in a ring of functions from the ground ring to itself: In addition to (pointwise) addition and multiplication, there is a compatible operation of composition.

Definition

A composition ring is a commutative ring $R$ equipped with an operation

$$(-)\circ(-) \colon R \times R \to R$$

such that for all elements $f \in R$, $g \in R$, and $h \in R$, we have:

1. $(f + g) \circ h = (f \circ h) + (g \circ h)$,

2. $(f \cdot g) \circ h = (f \circ h) \cdot (g \circ h)$,

3. $f \circ (g \circ h) = (f \circ g) \circ h$

   (associativity of composition).

Examples

• The endo]-[[function algebra on a commutative ring is a composition ring, with “$\circ$” being the composition of functions.

• In particular, every polynomial ring is a composition ring with “$\circ$” being the actual composition of polynomials regarded as functions from the ground ring to itself.

• Every commutative ring becomes a composition ring by setting $f \circ g \coloneqq f$.

See also

• commutative ring

• polynomial ring

• function algebra

• un-related: composition algebra

References

The concept is due to:

• Irving Adler, Composition rings, Duke Mathematical Journal, 29 4 (1962) 607–623, $[$doi:10.1215/S0012-7094-62-02961-7, ISSN 0012-7094, MR 0142573$]$

See also:

• Wikipedia, Composition ring

• Erhard Aichinger, The Structure of Composition Algebras(1998) $[$pdf$]$