Minimal primary decomposition

A minimal primary decomposition of an ideal

I

is an irredundant finite intersection

I = ⋂_{i} Q_{i}

of primary ideals whose radicals are pairwise distinct.

Table of contents

Associated prime of a primary decomposition ( ${Q_{i}}$ )

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For a finite minimal primary decomposition

I = ⋂_{i} Q_{i}

, these are the prime ideals

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

(I : x)

, where

x \in / I

and

(I : x)

is a colon ideal. Indeed,

(I : x) = ⋂_{i} (Q_{i} : x)

, and each factor is either the whole ring or a primary ideal with radical of an ideal

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 ⋂_{j \neq = i} Q_{j} ∖ Q_{i}

, giving

(I : x) = 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.

First uniqueness theorem for primary decomposition ( $Ass_{R} (R / I) = {Q_{i}}$ )

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For a commutative Noetherian ring, the distinct radicals in a minimal primary decomposition are exactly the associated primes of a module

R / I

, and hence are invariant. The injection into the sum of primary quotients proves one inclusion; an element supported on one component, supplied by irredundancy, proves the other. This does not imply uniqueness of embedded primary components.

Embedded primary component

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A component in a minimal primary decomposition is embedded if its radical of an ideal strictly contains another radical occurring in that decomposition. Such components need not be unique. For example,