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.


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 RR equipped with an operation

()()colonR×RR(-)\circ(-) \colonR \times R \to R

such that for all elements fRf \in R, gRg \in R, and hRh \in R, we have:

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

  2. (fg)h=(fh)(gh)(f \cdot g) \circ h = (f \circ h) \cdot (g \circ h),

  3. f(gh)=(fg)hf \circ (g \circ h) = (f \circ g) \circ h

    (associativity of composition).

Examples

See also

References

The concept is due to:

See also:

  • Wikipedia, Composition ring

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

Last revised on June 4, 2022 at 01:03:38. See the history of this page for a list of all contributions to it.