where the are primary in .
Not every ideal admits a primary decomposition, so we define a decomposable ideal to be one that does.
Example. Let and take . Then is decomposable, and a primary decomposition of is given by
since , , and are all primary ideals in .
Given a primary decomposition , we say that the decomposition is a minimal primary decomposition if for all , the prime ideals (where rad denotes the radical of an ideal) are distinct, and for all , we have
In the example above, the decomposition of is minimal, where as is not.
Every primary decomposition can be refined to admit a minimal primary decomposition.
|Date of creation||2013-03-22 14:15:05|
|Last modified on||2013-03-22 14:15:05|
|Last modified by||mathcam (2727)|
|Synonym||shortest primary decomposition|
|Defines||minimal primary decomposition|