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In abstract algebra, if I and J are ideals of a commutative ring R, their ideal quotient (I : J) is the set Then (I : J) is itself an ideal in R. The ideal quotient is viewed as a quotient because if and only if . The ideal quotient is useful for calculating primary decompositions. It also arises in the description of the set difference in algebraic geometry (see below). (I : J) is sometimes referred to as a colon ideal because of the notation. In the context of fractional ideals, there is a related notion of the inverse of a fractional ideal.

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  • Ideal quotient (en)
  • 아이디얼 몫 (ko)
  • イデアル商 (ja)
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  • In abstract algebra, if I and J are ideals of a commutative ring R, their ideal quotient (I : J) is the set Then (I : J) is itself an ideal in R. The ideal quotient is viewed as a quotient because if and only if . The ideal quotient is useful for calculating primary decompositions. It also arises in the description of the set difference in algebraic geometry (see below). (I : J) is sometimes referred to as a colon ideal because of the notation. In the context of fractional ideals, there is a related notion of the inverse of a fractional ideal. (en)
  • 가환대수학에서 아이디얼 몫(영어: ideal quotient)은 같은 가환환 속의 두 아이디얼에 대하여 정의되는 이항 연산이다. 이는 아이디얼에 대한, 나눗셈의 일반화이다. 대수기하학에서, 이는 두 부분 대수다양체의 ‘차집합’에 해당한다. (대수기하학에서 아이디얼의 곱셈은 대략 부분 대수다양체의 ‘합집합’에 해당하며, 이는 그 역연산에 ‘가장 가까운’ 연산이다.) (ko)
  • 抽象代数学において、I と J が可換環 R のイデアルのとき、それらの イデアル商（英: ideal quotient） I : J とは集合 である。これを (I : J) と書くこともある。すると I : J も R のイデアルである。イデアル商は商と見ることができる、なぜならば であることと であることが同値だからだ。例えば、整数環 Z において (6) : (3) = (2) が成り立つ。イデアル商は準素分解の計算に役立つ。また代数幾何において差集合の記述で現れる（下記参照）。 I : J はその表記により コロンイデアル（colon ideal）と呼ばれることがある。分数イデアルの文脈では、分数イデアルのインバースに関連した概念がある。 (ja)
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  • In abstract algebra, if I and J are ideals of a commutative ring R, their ideal quotient (I : J) is the set Then (I : J) is itself an ideal in R. The ideal quotient is viewed as a quotient because if and only if . The ideal quotient is useful for calculating primary decompositions. It also arises in the description of the set difference in algebraic geometry (see below). (I : J) is sometimes referred to as a colon ideal because of the notation. In the context of fractional ideals, there is a related notion of the inverse of a fractional ideal. (en)
  • 가환대수학에서 아이디얼 몫(영어: ideal quotient)은 같은 가환환 속의 두 아이디얼에 대하여 정의되는 이항 연산이다. 이는 아이디얼에 대한, 나눗셈의 일반화이다. 대수기하학에서, 이는 두 부분 대수다양체의 ‘차집합’에 해당한다. (대수기하학에서 아이디얼의 곱셈은 대략 부분 대수다양체의 ‘합집합’에 해당하며, 이는 그 역연산에 ‘가장 가까운’ 연산이다.) (ko)
  • 抽象代数学において、I と J が可換環 R のイデアルのとき、それらの イデアル商（英: ideal quotient） I : J とは集合 である。これを (I : J) と書くこともある。すると I : J も R のイデアルである。イデアル商は商と見ることができる、なぜならば であることと であることが同値だからだ。例えば、整数環 Z において (6) : (3) = (2) が成り立つ。イデアル商は準素分解の計算に役立つ。また代数幾何において差集合の記述で現れる（下記参照）。 I : J はその表記により コロンイデアル（colon ideal）と呼ばれることがある。分数イデアルの文脈では、分数イデアルのインバースに関連した概念がある。 (ja)
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