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.
symmetric monoidal (∞,1)-category of spectra
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.
A composition ring is a commutative ring $R$ equipped with an operation
such that for all elements $f \in R$, $g \in R$, and $h \in R$, we have:
$(f + g) \circ h = (f \circ h) + (g \circ h)$,
$(f \cdot g) \circ h = (f \circ h) \cdot (g \circ h)$,
$f \circ (g \circ h) = (f \circ g) \circ h$
(associativity of composition).
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 $f \circ g \coloneqq f$.
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 05:03:38. See the history of this page for a list of all contributions to it.