Last update: 23 December 2011
Let be a commutative ring.
- A prime ideal is an ideal of such that is an integral domain.
- A maximal ideal is an ideal of such that is a field.
Let be a ring.
- The nilradical of is the ideal
- The radical of an ideal is the ideal of corresponding to the ideal in .
Let be a commutative ring. Let be a prime ideal of .
- A primary ideal is an ideal of such that and every zero divisor in is nilpotent.
- A primary ideal is a primary ideal such that .
Let be a commutative ring and let be an ideal of .
- The ideal is primary if and only if for every such that either or .
- If is maximal then is primary.
- If is primary then is the minimal prime ideal containing .
is maximal then
has only one prime ideal (since
So every element of
is a unit or is nilpotent. So every zero divisor of
is nilpotent. So is primary.
Let be a commutative ring. An irreducible ideal is an ideal of such that
Let be a Noetherian ring.
- Every ideal is a finite intersection of irreducible ideals.
- Every irreducible ideal is primary.
a. If there is an ideal which is not a finite intersection of irreducible then, by Zorn's lemma, the set of ideals which are not a finite intersection of irreducibles has a maximal element . Then is reducible and
with and strictly larger than . So and are finite intersections of irreducibles. So is a finite intersection of irreducibles.
b. To show is primary, it is sufficient to show that
is primary in
Let us show that the ideal is primary in a Noetherian ring. Let
must stabilize and so
for large enough . Then
is irreducible and
Let be a commutative ring. Let be an ideal of .
- A primary decomposition of is
- A minimal primary decomposition is a primary decomposition of such that all
are primary ideals, all
are distinct, and, for all
- A decomposable ideal is an ideal with a primary decomposition.
- Any primary decomposition can be reduced to a minimal primary decomposition.
is a minimal primary decomposition of then
- The minimal primes in
are the minimal primes in
c. If is a prime ideal such that
for all then there exists
since is prime. So
for some . Thus, by part b., contains a minimal prime.
Notes and References
Where are these from?