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primary ideal (Definition)

An ideal $ Q$ in a commutative ring $ R$ is a primary ideal if for all elements $ x,y\in R$, we have that if $ xy\in Q$, then either $ x\in Q$ or $ y^n\in Q$ for some $ n\in\mathbb{N}$.

This is clearly a generalization of the notion of a prime ideal, and (very) loosely mirrors the relationship in $ \mathbb{Z}$ between prime numbers and prime powers.

It is clear that every prime ideal is primary.

Example. Let $ Q=(25)$ in $ R=\mathbb{Z}$. Suppose that $ xy\in Q$ but $ x\notin Q$. Then $ 25\vert xy$, but 25 does not divide $ x$. Thus 5 must divide $ y$, and thus some power of $ y$ (namely, $ y^2$), must be in $ Q$.

The radical of a primary ideal is always a prime ideal. If $ P$ is the radical of the primary ideal $ Q$, we say that $ Q$ is $ P$-primary.



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Also defines:  primary, $P$-primary
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Cross-references: radical, divide, clear, powers, prime, prime numbers, prime ideal, commutative ring, ideal
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This is version 3 of primary ideal, born on 2004-03-12, modified 2008-10-05.
Object id is 5697, canonical name is PrimaryIdeal.
Accessed 6101 times total.

Classification:
AMS MSC13C99 (Commutative rings and algebras :: Theory of modules and ideals :: Miscellaneous)

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