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  "body": "\n## Summary\n\nA near-ring is a set with an addition and a multiplication that satisfies most of the [[ring|0B1261B5ED]] axioms.\nAddition need not be commutative, and only one distributive law is required.\nThe standard example is the set of all maps from a [[group|G5S1M9]] to itself, with pointwise addition and composition.\nNear-rings stand to arbitrary maps of groups as rings stand to endomorphisms of abelian groups.\nThey appear in the study of nonlinear maps, of permutation groups and of combinatorial designs.\n\n## Explanations\n\n### L1 human\n\nFor an abelian group, its endomorphisms can be added and composed, and they form a ring.\nIf one takes all maps of the group to itself, not only homomorphisms, one can still add them pointwise and compose them.\nComposition then distributes over addition from one side only: $(f + g) \\circ h = f \\circ h + g \\circ h$ always holds, but $f \\circ (g + h) = f \\circ g + f \\circ h$ needs $f$ to respect addition.\nA near-ring is any structure with exactly the rules that survive.\n",
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