Near-ring

Remarks

A left near-ring asks instead $c \cdot (a + b) = c \cdot a + c \cdot b$; the two notions are mirror images. Every ring is a near-ring. Example: for a group $(G, +)$, let $M(G)$ be the set of all maps $G \to G$, with $(f + g)(x) = f(x) + g(x)$ and $f \cdot g = f \circ g$. Clause 1 holds pointwise, clause 2 because composition is associative, and clause 3 because $((f + g) \circ h)(x) = f(h(x)) + g(h(x))$. The other distributive law fails in $M(\Z)$: for the constant map $f = 1$, $f \circ (g + h) = 1$ but $f \circ g + f \circ h = 2$. Non-example: $\N = \set{0, 1, 2, \ldots}$ with the usual operations satisfies clauses 2 and 3, but fails clause 1, since $1$ has no additive inverse.