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In mathematics, more specifically abstract 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. 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. Many important results in the field of noncommutative algebra area apply to commutative rings as special cases.

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  • Noncommutative ring
  • 非可換環
  • Okommutativ ring
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  • 数学、特に現代代数学と環論において、非可換環(ひかかんかん、英: noncommutative ring)とは乗法が可換ではない環である。つまり、a•b ≠ b•a なる R の元 a, b が存在する。非可換環論 (noncommutative algebra) は可換とは限らない環に適用できる結果の研究であるが、この分野の多くの重要な結果は特別な場合として可換環にも適用できる。
  • Inom matematiken, speciellt inom abstrakta algebran och ringteori, är en okommutativ ring en ring vars multiplikation inte är kommutativ; i andra ord finns det element a och b av R med a·b ≠ b·a. Okommutativ algebra är studien av okommutativa ringar; många resultat i detta område gäller dock även för kommutativa ringar.
  • In mathematics, more specifically abstract 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. 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. Many important results in the field of noncommutative algebra area apply to commutative rings as special cases.
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