nLab composition ring

Contents

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.

Context

Algebra

higher algebra

universal algebra

Contents

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 $f \circ g \coloneqq f$.

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$]$

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