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In mathematics, more specifically modern algebra and ring theory, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exists ''a'' and ''b'' in ''R'' with ''a''·''b'' ≠ ''b''·''a''. Note that many authors use the term ''noncommutative ring'' to refer to rings which are not necessarily commutative, and hence include commutative rings in their definition. Noncommutative algebra is the study of results applying to rings that are not required to be commutative; however, many important results in this area apply to commutative rings as special cases. Although some authors don't assume their rings have a multiplicative identity, in this article we make that assumption unless stated otherwise. ==Examples== Some examples of rings which are not commutative follow: * the matrix ring of ''n''-by-''n'' matrices over the real numbers, where ''n''>1. * Hamilton's quaternions. * any group algebra made from a group that is not abelian. 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「Noncommutative ring」の詳細全文を読む スポンサード リンク
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