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In mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally from functions on groups. == Definition == A set ''N'' together with two binary operations + (called ''addition'') and ⋅ (called ''multiplication'') is called a (right) ''near-ring'' if: :A1: ''N'' is a group (not necessarily abelian) under addition; :A2: multiplication is associative (so ''N'' is a semigroup under multiplication); and :A3: multiplication distributes over addition on the ''right'': for any ''x'', ''y'', ''z'' in ''N'', it holds that (''x'' + ''y'')⋅''z'' = (''x''⋅''z'') + (''y''⋅''z'').〔G. Pilz, (1982), "Near-Rings: What They Are and What They Are Good For" in ''Contemp. Math.'', 9, pp. 97–119. Amer. Math. Soc., Providence, R.I., 1981.〕 Similarly, it is possible to define a ''left near-ring'' by replacing the right distributive law A3 by the corresponding left distributive law. However, near-rings are almost always written as right near-rings. An immediate consequence of this ''one-sided distributive law'' is that it is true that 0⋅''x'' = 0 but it is not necessarily true that ''x''⋅0 = 0 for any ''x'' in ''N''. Another immediate consequence is that (−''x'')⋅''y'' = −(''x''⋅''y'') for any ''x'', ''y'' in ''N'', but it is not necessary that ''x''⋅(−''y'') = −(''x''⋅''y''). A near-ring is a ring (not necessarily with unity) if and only if addition is commutative and multiplication is distributive over addition on the ''left''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Near-ring」の詳細全文を読む スポンサード リンク
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