Composition ring
In mathematics, a composition ring, introduced in (Adler 1962), is a commutative ring (R, 0, +, −, ·), possibly without an identity 1, together with an operation ∘ : R × R → R {\displaystyle \circ :R\times R\rightarrow R} such that, for any three elements f , g , h ∈ R {\displaystyle f,g,h\in R} one has ( f + g ) ∘ h = ( f ∘ h ) + ( g ∘ h ) {\displaystyle (f+g)\circ h=(f\circ h)+(g\circ h)} ( f ⋅ g ) ∘ h = ( f ∘ h ) ⋅ ( g ∘ h ) {\displaystyle (f\cdot g)\circ h=(f\circ h)\cdot (g\circ h)} ( f ∘ g ) ∘ h = f ∘ ( g ∘ h ) . {\displaystyle (f\circ g)\circ h=f\circ (g\circ h).} It is not generally the case that f ∘ g = g ∘ f {\displaystyle f\circ g=g\circ f} , nor is it generally the case that f ∘ ( g + h ) {\displaystyle f\circ (g+h)} (or f ∘ ( g ⋅ h ) {\displaystyle f\circ (g\cdot h)} ) has any algebraic relationship to f ∘ g {\displaystyle f\circ g} and f ∘ h {\displaystyle f\circ h} .