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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 $f\circ h$ .