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  • ...[[ring]] <math>R</math> is '''prime''' if <math>P\ne R</math> and for any ideals <math>I,J\subseteq R</math> with <math>IJ\subseteq P</math> we have either ...nd definition easily implies the the following important property of prime ideals in commutative rings with unity:
    967 bytes (176 words) - 17:08, 7 April 2012
  • ...[antichain]]s in [[poset]]s. Specifically, it ask for the number of order ideals of the [[product poset]] of the [[chain]] of length <math>4</math> and the
    2 KB (443 words) - 21:41, 22 December 2021
  • ==Invertibility of Ideals== '''Theorem:''' All fractional ideals of <math>R</math> are invertible.
    9 KB (1,648 words) - 15:36, 14 October 2017
  • ...ings are the [[kernel]]s of ring [[homomorphism]]s; in this way, two-sided ideals of rings are similar to [[normal subgroup]]s of [[group]]s. ...Note that the right ideals of a ring <math>A</math> are exactly the left ideals of the opposite ring <math>A^0</math>.
    8 KB (1,389 words) - 22:44, 17 February 2020
  • 608 bytes (89 words) - 15:05, 13 June 2008
  • ...does not contain 1. Let <math>\mathcal{F}</math> be the family of proper ideals of <math>R</math> containig <math>\mathfrak{a}</math>. Evidently, 1 is not
    1 KB (192 words) - 07:38, 16 June 2008
  • ''Proof.'' We prove the case for left ideals; the other case follows from passing to the opposite ring.
    3 KB (547 words) - 13:13, 16 June 2008
  • ...of <math>A</math> that is not nilpotent. Let <math>S</math> be the set of ideals of <math>A</math> that do not contain any element of the form <math>a^n</ma
    3 KB (532 words) - 11:14, 20 September 2008
  • ...her general question in commutative algebra: What is the behavior of prime ideals under ring homomorphisms? This theorem resolves this question for the cano .../math> becomes <math>(a+bi)(a-bi)</math>, for some <math>a,b</math>; these ideals are then prime.
    4 KB (612 words) - 20:31, 7 January 2025
  • One important property of maximal ideals is that if <math>R</math> is a [[commutative ring]] with unity then the [[q
    968 bytes (183 words) - 18:50, 23 August 2009
  • \mathfrak{c}_{M,0}</math>, and these ideals are the sets of degree-zero
    4 KB (617 words) - 18:59, 23 April 2023
  • [[ideal | ideals]]/[[submodules]]. The notion is similar to that
    2 KB (263 words) - 20:00, 19 April 2012
  • 6 KB (1,217 words) - 22:05, 23 August 2009
  • ...ion, regular [[ideal]]s of <math>R</math> are sometimes called '''integral ideals'''. Clearly any integral ideal of <math>R</math> is a fractional ideal of < ...ow be defined just as they are for integral ideals. Namely, for fractional ideals <math>I</math> and <math>J</math> we let <math>I+J</math> be the submodule
    2 KB (288 words) - 19:05, 23 January 2017
  • ...the case, the concept can be largely recovered through the use of [[ideal|ideals]]. The general idea is to consider entities that "divide" irreducibles, for we define ideals <math>\mathfrak{a}, \mathfrak{b}, \mathfrak{c}, \mathfrak{d}</math> such th
    10 KB (1,646 words) - 14:04, 28 May 2020
  • ...it’s Windows 10’s default. We’ve as of late run pieces praising the ideals of Firefox, Vivaldi, Edge, and Drama to mess with your choices.
    7 KB (1,112 words) - 11:38, 27 September 2024