Prime avoidance lemma: Difference between revisions

Revision as of 18:20, 17 December 2007

Statement

Let $R$ be a commutative unital ring. Let $I_{1}, I_{2}, \dots, I_{n}$ and $J$ be ideals of $R$ , such that $J \subset ⋃_{j} I_{j}$ . Then, if $R$ contains an infinite field or if at most two of the $I_{j}$ s are not prime, then $J$ is contained in one of the $I_{j}$ s.

Graded version

If $R$ is graded, and $J$ is generated by homogeneous elements of positive degree, then it suffices to assume that the homogeneous elements of $J$ are contained in $⋃_{j} I_{j}$ .

Importance

The prime avoidance lemma is useful for establishing dichotomies; in particular, if $J$ is an ideal which is not cintained in any of the $I_{j}$ s, then $J$ has an element which is contained in none of the $I_{j}$ s.

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 ==Statement==
-Let <math>R</math> be a [[commutative unital ring]]. Let <math>I_1, I_2, \ldots, I_n</math> and <math>J</math> be [[ideal]]s of <math>R</math>, such that <math>J \subset \bigcup_j I_j</math>. Then, if <math>R</math> contains an infinite field or if at most two of the <math>I_j</math>s are [[prime ideal|prime]], then <math>J</math> is contained in one of the <math>I_j</math>s.
+Let <math>R</math> be a [[commutative unital ring]]. Let <math>I_1, I_2, \ldots, I_n</math> and <math>J</math> be [[ideal]]s of <math>R</math>, such that <math>J \subset \bigcup_j I_j</math>. Then, if <math>R</math> contains an infinite field or if at most two of the <math>I_j</math>s are not [[prime ideal|prime]], then <math>J</math> is contained in one of the <math>I_j</math>s.
 ===Graded version===
 If <math>R</math> is [[graded ring|graded]], and <math>J</math> is generated by homogeneous elements of positive degree, then it suffices to assume that the homogeneous elements of <math>J</math> are contained in <math>\bigcup_j I_j</math>.
+==Importance==
+The prime avoidance lemma is useful for establishing dichotomies; in particular, if <math>J</math> is an ideal which is not cintained in any of the <math>I_j</math>s, then <math>J</math> has an element which is contained in ''none'' of the <math>I_j</math>s.