Composition ring information

Algebraic structure → Ring theory Ring theory
Basic concepts Rings • Subrings • Ideal • Quotient ring • Fractional ideal • Total ring of fractions • Product of rings • Free product of associative algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative algebra • Graded ring • Involutive ring • Category of rings • Initial ring $\mathbb {Z}$ • Terminal ring $0=\mathbb {Z} /1\mathbb {Z}$ Related structures • Field • Finite field • Non-associative ring • Lie ring • Jordan ring • Semiring • Semifield
Commutative algebra Commutative rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite field • Composition ring • Polynomial ring • Formal power series ring Algebraic number theory • Algebraic number field • Ring of integers • Algebraic independence • Transcendental number theory • Transcendence degree p-adic number theory and decimals • Direct limit/Inverse limit • Zero ring $\mathbb {Z} /1\mathbb {Z}$ • Integers modulo pⁿ $\mathbb {Z} /p^{n}\mathbb {Z}$ • Prüfer p-ring $\mathbb {Z} (p^{\infty })$ • Base-p circle ring $\mathbb {T}$ • Base-p integers $\mathbb {Z}$ • p-adic rationals $\mathbb {Z} [1/p]$ • Base-p real numbers $\mathbb {R}$ • p-adic integers $\mathbb {Z} _{p}$ • p-adic numbers $\mathbb {Q} _{p}$ • p-adic solenoid $\mathbb {T} _{p}$ Algebraic geometry • Affine variety
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In mathematics, a composition ring, introduced in (Adler 1962), 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

such that, for any three elements $f,g,h\in R$ one has

$(f+g)\circ h=(f\circ h)+(g\circ h)$
$(f\cdot g)\circ h=(f\circ h)\cdot (g\circ h)$
$(f\circ g)\circ h=f\circ (g\circ h).$

It is not generally the case that $f\circ g=g\circ f$ , nor is it generally the case that $f\circ (g+h)$ (or $f\circ (g\cdot h)$ ) has any algebraic relationship to $f\circ g$ and $f\circ h$ .

Composition ring information

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Composition ring

Function composition

Chiastic structure

Ring homomorphism

Composition operator

Commutative ring

Semiring

Ring theory

Polynomial ring

Quotient ring

Zero ring

Composition series

Associative algebra

Formal power series

Ring of integers

Integral domain

Integer

Noncommutative ring

Commutative algebra

Semiprimitive ring

Field of fractions

Endomorphism ring