Editing Primary decomposition (section)

=== 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.