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An ideal  in a commutative ring  is a primary ideal if for all elements  , we have that if  , then either  or  for some
 .
This is clearly a generalization of the notion of a prime ideal, and (very) loosely mirrors the relationship in
 between prime numbers and prime powers.
It is clear that every prime ideal is primary.
Example. Let  in
 . Suppose that  but  . Then  , but 25 does not divide  . Thus 5 must divide  , and thus some power of  (namely,  ), must be in  .
The radical of a primary ideal is always a prime ideal. If  is the radical of the primary ideal  , we say that  is  -primary.
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