primary decomposition theorem
Every decomposable ideal in a commutative ring with 1 has a unique minimal primary decomposition. In other words, if is an ideal of with two minimal primary decompositions
then , and after some rearrangement, .
The theorem says, that, the number of primary components of a minimal primary decomposition of an ideal, as well as the set of prime radicals associated with the primary components, are unique. This is not to say, however, that the ideal has a unique minimal primary decomposition. For example, let be a field. Consider the ring of polynomials over in two variables. The ideal has minimal primary decompositions for every .
where each is a prime number. This is the same as saying that
- 1 D.G. Northcott, Ideal Theory, Cambridge University Press, 1953.
|Title||primary decomposition theorem|
|Date of creation||2013-03-22 18:19:56|
|Last modified on||2013-03-22 18:19:56|
|Last modified by||CWoo (3771)|