Editing Commutative algebra (section)

===Primary decomposition===
{{Main|Primary decomposition}}
An ideal ''Q'' of a ring is said to be ''[[Primary ideal|primary]]'' if ''Q'' is [[proper subset|proper]] and whenever ''xy'' ∈ ''Q'', either ''x'' ∈ ''Q'' or ''y<sup>n</sup>'' ∈ ''Q'' for some positive integer ''n''. In '''Z''', the primary ideals are precisely the ideals of the form (''p<sup>e</sup>'') where ''p'' is prime and ''e'' is a positive integer. Thus, a primary decomposition of (''n'') corresponds to representing (''n'') as the intersection of finitely many primary ideals.

The ''[[Lasker–Noether theorem]]'', given here, may be seen as a certain generalization of the fundamental theorem of arithmetic:

{{math theorem|name=Lasker-Noether Theorem
|math_statement=Let ''R'' be a commutative Noetherian ring and let ''I'' be an ideal of ''R''. Then ''I'' may be written as the intersection of finitely many primary ideals with distinct [[Radical of an ideal|radicals]]; that is:

: <math>I=\bigcap_{i=1}^t Q_i</math>

with ''Q<sub>i</sub>'' primary for all ''i'' and Rad(''Q<sub>i</sub>'') ≠ Rad(''Q<sub>j</sub>'') for ''i'' ≠ ''j''. Furthermore, if:

: <math>I=\bigcap_{i=1}^k P_i</math>

is decomposition of ''I'' with Rad(''P<sub>i</sub>'') ≠ Rad(''P<sub>j</sub>'') for ''i'' ≠ ''j'', and both decompositions of ''I'' are ''irredundant'' (meaning that no proper subset of either {''Q''<sub>1</sub>, ..., ''Q<sub>t</sub>''} or {''P''<sub>1</sub>, ..., ''P<sub>k</sub>''} yields an intersection equal to ''I''), ''t'' = ''k'' and (after possibly renumbering the ''Q<sub>i</sub>'') Rad(''Q<sub>i</sub>'') = Rad(''P<sub>i</sub>'') for all ''i''.}}

For any primary decomposition of ''I'', the set of all radicals, that is, the set {Rad(''Q''<sub>1</sub>), ..., Rad(''Q<sub>t</sub>'')} remains the same by the Lasker–Noether theorem. In fact, it turns out that (for a Noetherian ring) the set is precisely the [[associated prime|assassinator]] of the module ''R''/''I''; that is, the set of all [[annihilator (ring theory)|annihilators]] of ''R''/''I'' (viewed as a module over ''R'') that are prime.