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Let  be a commutative ring and  be an ideal in  . A primary decomposition of  is a way of writing  as a finite intersection of primary ideals:
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
 rad (where rad denotes the radical of an ideal) are distinct, and for all
 , we have
Every primary decomposition can be refined to admit a minimal primary decomposition.
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