Editing Primary decomposition (section)

== Non-Noetherian case ==
The next theorem gives necessary and sufficient conditions for a ring to have primary decompositions for its ideals.
{{math_theorem|math_statement=Let ''R'' be a commutative ring. Then the following are equivalent.
# Every ideal in ''R'' has a primary decomposition.
# ''R'' has the following properties:
#*(L1) For every proper ideal ''I'' and a prime ideal ''P'', there exists an ''x'' in ''R'' - ''P'' such that (''I'' : ''x'') is the pre-image of ''I'' ''R''<sub>''P''</sub> under the localization map ''R'' → ''R''<sub>''P''</sub>.
#*(L2) For every ideal ''I'', the set of all pre-images of ''I'' ''S''<sup>−1</sup>''R'' under the localization map ''R'' → ''S''<sup>−1</sup>''R'', ''S'' running over all multiplicatively closed subsets of ''R'', is finite.}}

The proof is given at Chapter 4 of Atiyah–Macdonald as a series of exercises.<ref>{{harvnb|Atiyah|Macdonald|1994}}</ref>

There is the following uniqueness theorem for an ideal having a primary decomposition.

{{math_theorem|Let ''R'' be a commutative ring and ''I'' an ideal. Suppose ''I'' has a minimal primary decomposition <math>I = \cap_1^r Q_i</math> (note: "minimal" implies <math>\sqrt{Q_i}</math> are distinct.) Then
# The set <math>E = \left \{ \sqrt{Q_i} | 1 \le i \le r \right \}</math> is the set of all prime ideals in the set <math>\left\{ \sqrt{(I : x)} | x \in R \right\}</math>.
# The set of minimal elements of ''E'' is the same as the set of [[minimal prime ideal]]s over ''I''. Moreover, the primary ideal corresponding to a minimal prime ''P'' is the pre-image of ''I'' ''R''<sub>''P''</sub> and thus is uniquely determined by ''I''.}}

Now, for any commutative ring ''R'',  an ideal ''I'' and a minimal prime ''P'' over ''I'', the pre-image of ''I'' ''R''<sub>''P''</sub> under the localization map is the smallest ''P''-primary ideal containing ''I''.<ref>{{harvnb|Atiyah|Macdonald|1994|loc=Ch. 4. Exercise 11}}</ref> Thus, in the setting of preceding theorem, the primary ideal ''Q'' corresponding to a minimal prime ''P'' is also the smallest ''P''-primary ideal containing ''I'' and is called the ''P''-primary component of ''I''.

For example, if the power ''P''<sup>''n''</sup> of a prime ''P'' has a primary decomposition, then its ''P''-primary component is the ''n''-th [[symbolic power of a prime ideal|symbolic power]] of ''P''.