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# primary ideal

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.

Example. Let $Q=(25)$ in $R=\mathbb{Z}$. Suppose that $xy\in Q$ but $x\notin Q$. Then $25|xy$, 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*.

Defines:

primary, $P$-primary

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Definition

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## Mathematics Subject Classification

13C99*no label found*

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