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A near-ring is a ring if and only if multiplication also distributes over addition on the left. (It follows then that addition is commutative).

This is not true! Take a nonabelian group (G, +) and define the multiplication trivially, i.e. x•y = 0 for all x,y ε G. Then • is commutative and distributive over +, but (G, +, •) is not a ring. 00:47, 11 April 2007 (UTC)