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radical ideal

名詞

radical ideal (複数形 radical ideals)

(algebra, commutative algebra, ring theory) An ideal I within a ring R that is its own radical (i.e., for any r ∈ R, if rn ∈ I for some positive integer n, then r ∈ I).
- 1981, Harry C. Hutchins, Examples of Commutative Rings, Polygonal Publishing House, page 3,
  Any intersection of radical ideals is again a radical ideal. Since not every intersection of prime ideals is a prime ideal, it follows that not all radical ideals are prime.
- 1995, David Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, Springer, page 33,
  It follows at once that R / I {\displaystyle R/I} $R/I$ is a reduced ring iff I {\displaystyle I} $I$ is a radical ideal. Thus, the ideals I ( X ) {\displaystyle I(X)} $I(X)$ are all radical ideals.
- 1997, D. D. Anderson, Dong Je Kwak, Some Remarks on G-Noetherian Rings, Paul-Jean Cahen, Marco Fontana, Evan Houston, Salah-Eddine Kabbaj (editors), Commutative Ring Theory: Proceedings of the II International Conference, Marcel Dekker, page 30,
  Thus Theorem 2 [8] is a special case of the well-known result that a commutative ring R satisfies the ascending chain condition on radical ideals if and only if R satisfies the ascending chain condition on prime ideals and each radical ideal of R is a finite intersection of prime ideals […] .

同意語

(ideal that is its own radical): semiprime ideal

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