Editing Primary decomposition (section)

===Examples===
The examples of the section are designed for illustrating some properties of primary decompositions, which may appear as surprising or counter-intuitive. All examples are ideals in a [[polynomial ring]] over a [[field (mathematics)|field]] {{math|''k''}}.

====Intersection vs. product====
The primary decomposition in <math>k[x,y,z]</math> of the ideal <math>I=\langle x,yz \rangle</math> is
:<math>I = \langle x,yz \rangle = \langle x,y \rangle \cap \langle x,z \rangle.</math>

Because of the generator of degree one, {{math|''I''}} is not the product of two larger ideals. A similar example is given, in two indeterminates by
:<math>I = \langle x,y(y+1) \rangle = \langle x,y \rangle \cap \langle x,y+1 \rangle.</math>

====Primary vs. prime power====
In <math>k[x,y]</math>, the ideal <math>\langle x,y^2 \rangle</math> is a primary ideal that has <math>\langle x,y \rangle</math> as associated prime. It is not a power of its associated prime.

====Non-uniqueness and embedded prime ====

For every positive integer {{math|''n''}}, a primary decomposition in <math>k[x,y]</math> of the ideal <math>I=\langle x^2, xy \rangle</math> is
:<math>I = \langle x^2,xy \rangle = \langle x \rangle \cap \langle x^2, xy, y^n \rangle.</math>

The associated primes are 
:<math>\langle x \rangle \subset \langle x,y \rangle.</math>

Example: Let ''N''&nbsp;=&nbsp;''R''&nbsp;=&nbsp;''k''[''x'',&nbsp;''y''] for some field ''k'', and let ''M'' be the ideal (''xy'',&nbsp;''y''<sup>2</sup>). Then ''M'' has two different minimal primary decompositions
''M'' = (''y'') &cap; (''x'', ''y''<sup>2</sup>) = (''y'') &cap; (''x''&nbsp;+&nbsp;''y'',&nbsp;''y''<sup>2</sup>).
The minimal prime is (''y'') and the embedded prime is (''x'',&nbsp;''y'').

====Non-associated prime between two associated primes====
In <math>k[x,y,z],</math> the ideal <math>I=\langle x^2, xy, xz \rangle</math> has the (non-unique) primary decomposition
:<math>I = \langle x^2,xy, xz \rangle = \langle x \rangle \cap \langle x^2, y^2, z^2, xy, xz, yz \rangle.</math>
The associated prime ideals are <math>\langle x \rangle \subset \langle x,y,z \rangle,</math> and <math>\langle x, y \rangle</math> is a non associated prime ideal such that
:<math>\langle x \rangle \subset \langle x,y \rangle \subset \langle x,y,z \rangle.</math>
====A complicated example====
Unless for very simple examples, a primary decomposition may be hard to compute and may have a very complicated output. The following example has been designed for providing such a complicated output, and, nevertheless, being accessible to hand-written computation.

Let
:<math>
\begin {align}
P&=a_0x^m + a_1x^{m-1}y +\cdots +a_my^m \\
Q&=b_0x^n + b_1x^{n-1}y +\cdots +b_ny^n
\end {align}</math>
be two [[homogeneous polynomial]]s in {{math|''x'', ''y''}}, whose coefficients <math>a_1, \ldots, a_m, b_0, \ldots, b_n</math> are polynomials in other indeterminates <math>z_1, \ldots, z_h</math> over a field {{math|''k''}}. That is, {{math|''P''}} and {{math|''Q''}} belong to <math>R=k[x,y,z_1, \ldots, z_h],</math> and it is in this ring that a primary decomposition of the ideal <math>I=\langle P,Q\rangle</math> is searched. For computing the primary decomposition, we suppose first that 1 is a [[Polynomial greatest common divisor|greatest common divisor]] of {{math|''P''}} and {{math|''Q''}}.

This condition implies that {{math|''I''}} has no primary component of [[height (ring theory)|height]] one. As {{math|''I''}} is generated by two elements, this implies that it is a [[complete intersection]] (more precisely, it defines an [[algebraic set]], which is a complete intersection), and thus all primary components have height two. Therefore, the associated primes of {{math|''I''}} are exactly the primes ideals of height two that contain {{math|''I''}}.

It follows that <math>\langle x,y\rangle</math> is an associated prime of {{math|''I''}}.

Let <math>D\in k[z_1, \ldots, z_h]</math> be the [[Resultant#Homogeneous resultant|homogeneous resultant]] in {{math|''x'', ''y''}} of {{math|''P''}} and {{math|''Q''}}. As the greatest common divisor of {{math|''P''}} and {{math|''Q''}} is a constant, the resultant {{math|''D''}} is not zero, and resultant theory implies that {{math|''I''}} contains all products of {{math|''D''}} by a [[monomial]] in {{math|''x'', ''y''}} of degree {{math|''m'' + ''n'' – 1}}. As <math>D\not\in \langle x,y\rangle,</math> all these monomials belong to the primary component contained in <math>\langle x,y\rangle.</math> This primary component contains {{math|''P''}} and {{math|''Q''}}, and the behavior of primary decompositions under [[localization of a ring|localization]] shows that this primary component is 
:<math>\{t|\exists e, D^et \in I\}.</math>

In short, we have a primary component, with the very simple associated prime <math>\langle x,y\rangle,</math> such all its generating sets involve all indeterminates.

The other primary component contains {{math|''D''}}. One may prove that if {{math|''P''}} and {{math|''Q''}} are sufficiently [[generic property|generic]] (for example if the coefficients of {{math|''P''}} and {{math|''Q''}} are distinct indeterminates), then there is only another primary component, which is a prime ideal, and is generated by {{math|''P''}}, {{math|''Q''}} and {{math|''D''}}.