Primary ideal

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In mathematics, an ideal <math>Q</math> in a commutative ring <math>R</math> is a primary ideal if for all elements <math>x,y\in R</math>, we have that if <math>xy\in Q</math>, then either <math>x\in Q</math> or <math>y^n\in Q</math> for some <math>n\in\mathbb{N}.</math>

Alternatively, an ideal <math>\mathfrak{q}</math> is primary if and only if <math>A/\mathfrak{q}\neq 0 </math> and every zero divisor is nilpotent.

This is a generalization of the notion of a prime ideal (thus every prime ideal is primary), and (very) loosely mirrors the relationship in <math>\mathbb{Z}</math> between prime numbers and prime powers.

If the radical of the primary ideal <math>Q</math> is the prime ideal <math>P</math>, then <math>Q</math> is said to be <math>P</math>-primary.

[edit] Example

Let <math>Q=(125)</math> in <math>R=\mathbb{Z}.</math> Suppose that <math>xy\in Q</math> but <math>x\notin Q.</math> Then <math>125| xy</math>, but 125 does not divide <math>x.</math> Thus 5 must divide <math>y</math>, so some power of <math>y</math> (namely, <math>y^3</math>), must be in <math>Q.</math> Therefore <math>Q</math> is primary.