----- Let $R$ be a ([[commutative ring|commutative]]) [[ring]], $I \subset R$ an [[ideal]]. > [!proposition] Proposition. ([[uniqueness properties of primary decomposition]]) > If a [[primary ideal|primary decomposition]] of $I$ exists, it need not be unique. That said, the following notions depend only on $I$ and not on the choice of [[primary ideal|primary decomposition]]. Suppose $I$ has a minimal[^1] primary decomposition $I=\mathfrak{q}_{1} \cap \dots \cap \mathfrak{q}_{n}$, and write $\mathfrak{p}_{i}=\sqrt{ \mathfrak{q}_{i} }$. > > **The associated prime ideals of $I$.** $\mathfrak{p}_{1}, \dots, \mathfrak{p}_{n}$ depend only on $I$. In fact, $\{ \mathfrak{p}_{1},\dots, \mathfrak{p}_{n} \}=\{ (I : x) : x \in R \} \cap \text{Spec } R.$ > (The RHS clearly only depends on $I$.)[^2] > > **The isolated prime ideals of $I$.** The set of minimal elements among $\mathfrak{p}_{1},\dots,\mathfrak{p}_{n}$ consists exactly of the minimal prime ideals of $R$ over $I$; equivalently, the [[prime ideal|prime ideals]] of $R$ [[the correspondence theorem for rings|corresponding]] to the [[minimal prime ideal|minimal prime ideals]] of $R / I$. > > **The embedded prime ideals of $I$.** An associated prime ideal is said to be **embedded** if it is not isolated. > > **The isolated primary components of $I$.** If $\mathfrak{p}_{1},\dots,\mathfrak{p}_{t}$, $t \leq n$, are the isolated [[prime ideal|prime ideals]] of $I$, then $\mathfrak{q}_{1},\dots,\mathfrak{q}_{t}$ depend only on $I$. In fact, $\mathfrak{q}_i=I^{ec}$ wrt the [[localization|localization map]] $R \to R_{\mathfrak{p}_{i}}$. Note that if $t=n$, meaning associated $\iff$ isolated for $I$, then this means the primary decomposition is unique! This happens e.g. when $I=\sqrt{ I }$ is [[radical of an ideal|radical]]. ----- #### [^1]: Recall that any primary decomposition can be made minimal. [^2]: Recall the notation $(I:x)$ from [[ideal quotient]]. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```