Editing Primary decomposition (section)

==Primary decomposition of an ideal==
Let <math>R</math> be a Noetherian commutative ring. An ideal <math>I</math> of <math>R</math> is called '''[[primary ideal|primary]]''' if it is a [[proper ideal]] and for each pair of elements <math>x</math> and <math>y</math> in <math>R</math> such that <math>xy</math> is in <math>I</math>, either <math>x</math> or some power of <math>y</math> is in <math>I</math>; equivalently, every [[zero-divisor]] in the [[quotient ring|quotient]] <math>R/I</math> is nilpotent. The [[radical of an ideal|radical]] of a primary ideal <math>Q</math> is a prime ideal and <math>Q</math> is said to be <math>\mathfrak{p}</math>-primary for <math>\mathfrak{p} = \sqrt{Q}</math>.

Let <math>I</math> be an ideal in <math>R</math>. Then <math>I</math> has an '''irredundant primary decomposition''' into primary ideals:

:<math>I = Q_1 \cap \cdots \cap Q_n\ </math>.

Irredundancy means:

*Removing any of the <math>Q_i</math> changes the intersection, i.e. for each <math>i</math> we have: <math>\cap_{j \ne i} Q_j \not\subset Q_i</math>.
*The [[prime ideal]]s <math>\sqrt{Q_i}</math> are all distinct.

Moreover, this decomposition is unique in the two ways:
*The set <math>\{ \sqrt{Q_i} \mid i \}</math> is uniquely determined by <math>I</math>, and
*If <math>\mathfrak{p} = \sqrt{Q_i}</math> is a minimal element of the above set, then <math>Q_i</math> is uniquely determined by <math>I</math>; in fact, <math>Q_i</math> is the pre-image of <math>I R_{\mathfrak{p}}</math> under the [[localization map]] <math>R \to R_{\mathfrak{p}}</math>.
Primary ideals which correspond to non-minimal prime ideals over <math>I</math> are in general not unique (see an example below). For the existence of the decomposition, see [[#Primary decomposition from associated primes]] below.

The elements of <math>\{ \sqrt{Q_i} \mid i \}</math> are called the '''prime divisors''' of <math>I</math> or the '''primes belonging to''' <math>I</math>. In the language of module theory, as discussed below, the set <math>\{ \sqrt{Q_i} \mid i \}</math> is also the set of associated primes of the <math>R</math>-module <math>R/I</math>. Explicitly, that means that there exist elements <math>g_1, \dots, g_n</math> in <math>R</math> such that
:<math>\sqrt{Q_i} = \{ f \in R \mid fg_i \in I \}.</math><ref>In other words, <math>\sqrt{Q_i} = (I : g_i)</math> is the ideal quotient.</ref>

By a way of shortcut, some authors call an associated prime of <math>R/I</math> simply an associated prime of <math>I</math> (note this practice will conflict with the usage in the module theory).
*The minimal elements of <math>\{ \sqrt{Q_i} \mid i \}</math> are the same as the [[minimal prime ideal]]s containing <math>I</math> and are called '''isolated primes'''.
*The non-minimal elements, on the other hand, are called the '''embedded primes'''.

In the case of the ring of integers <math>\mathbb Z</math>, the Lasker–Noether theorem is equivalent to the [[fundamental theorem of arithmetic]]. If an integer <math>n</math> has prime factorization <math>n = \pm p_1^{d_1} \cdots p_r^{d_r}</math>, then the primary decomposition of the ideal <math>\langle n \rangle</math> generated by <math>n</math> in <math>\mathbb Z</math>, is

:<math>\langle n\rangle = \langle p_1^{d_1} \rangle \cap \cdots \cap \langle p_r^{d_r}\rangle.</math>

Similarly, in a [[unique factorization domain]], if an element has a prime factorization <math>f = u p_1^{d_1} \cdots p_r^{d_r},</math> where <math>u</math> is a [[unit (ring theory)|unit]], then the primary decomposition of the [[principal ideal]] generated by <math>f</math> is 
:<math>\langle f\rangle = \langle p_1^{d_1} \rangle \cap \cdots \cap \langle p_r^{d_r}\rangle.</math>

===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''}}.

=== Geometric interpretation ===
In [[algebraic geometry]], an [[affine algebraic set]] {{math|''V''(''I'')}} is defined as the set of the common [[zero of a function|zeros]] of an ideal {{math|''I''}} of a [[polynomial ring]] <math>R=k[x_1,\ldots, x_n].</math>

An irredundant primary decomposition
:<math>I=Q_1\cap\cdots\cap Q_r</math>
of {{math|''I''}} defines a decomposition of {{math|''V''(''I'')}} into a union of algebraic sets {{math|''V''(''Q''<sub>''i''</sub>)}}, which are irreducible, as not being the union of two smaller algebraic sets.

If <math>P_i</math> is the [[associated prime]] of <math>Q_i</math>, then <math>V(P_i)=V(Q_i),</math> and Lasker–Noether theorem shows that {{math|''V''(''I'')}} has a unique irredundant decomposition into irreducible [[algebraic varieties]]
:<math>V(I)=\bigcup V(P_i),</math> 
where the union is restricted to minimal associated primes. These minimal associated primes are the primary components of the [[radical of an ideal|radical]] of {{math|''I''}}. For this reason, the primary decomposition of the radical of {{math|''I''}} is sometimes called the ''prime decomposition'' of {{math|''I''}}.

The components of a primary decomposition (as well as of the algebraic set decomposition) corresponding to minimal primes are said ''isolated'', and the others are said ''{{vanchor|embedded}}''.

For the decomposition of algebraic varieties, only the minimal primes are interesting, but in [[intersection theory]], and, more generally in [[scheme theory]], the complete primary decomposition has a geometric meaning.