## Primary Ideals, Primary Decomposition

### Primary Decomposition

An ideal J has a primary decomposition if it is the intersection of a finite number
of primary ideals.
If no ideal in the list contains the intersection of the others,
and all ideals have distinct prime radicals, the decomposition is reduced.
Every primary decomposition implies a reduced primary decomposition, as follows.

Let P be the radical ideal of one of the primary ideals in the list.
Let G be the intersection of all the P primary ideals in the list.
As we saw in the previous section, G is primary, and rad(G) = P.
Replace all the P primary ideals with G.
Do this for each prime radical, and each P is represented only once.

Finally, if one of the primary ideals contains the intersection of the others, remove it, since it does not change J.
Repeat this step until you have a reduced decomposition.