= Associated prime of a primary decomposition
{title2=$\{\sqrt{Q_i}\}$}
= Associated primes of a primary decomposition
{synonym}
For a finite <minimal primary decomposition> $I=\bigcap_iQ_i$, these are the <prime ideals> $\sqrt{Q_i}$. This terminology applies without a <Noetherian> hypothesis when the decomposition exists. Its invariance follows from the intrinsic description as those <prime ideals> of the form $\sqrt{(I:x)}$, where $x\notin I$ and $(I:x)$ is a <colon ideal>. Indeed, $(I:x)=\bigcap_i(Q_i:x)$, and each factor is either the whole <ring> or a <primary ideal> with <radical of an ideal> $\sqrt{Q_i}$. A finite intersection of <prime ideals> can be prime only if it equals one of them: otherwise multiplying one element outside the intersection from each factor contradicts the <prime ideal> condition. Conversely, irredundancy supplies $x\in\bigcap_{j\ne i}Q_j\setminus Q_i$, giving $\sqrt{(I:x)}=\sqrt{Q_i}$. In a <Noetherian ring> these agree with the <associated primes of a module> $R/I$. For arbitrary <rings>, the <annihilator> definition of an <associated prime of a module> must not be substituted without a further hypothesis.
\Comment[[
Exact target: maximal-primary-ideal-criterion
Existing parent: Primary ideal
Intended file: /home/ciro/bak/git/codex-wiki/commutative-algebra.bigb
]]
Back to article page
