1. ## Primary decomposition

Let $\displaystyle P_1=<x,y>, P_2=<x,z>, P_3=<y,z> \in \mathbb{Q}[x,y,z]$.
Compute an irredundant primary decomposition and the associated primes of the product $\displaystyle P_1P_2P_3$.

How can I compute a primary decomposition???
If you serve solution of this problem, then it's very helpful for me.

2. Originally Posted by Stiger
Let $\displaystyle P_1=<x,y>, P_2=<x,z>, P_3=<y,z> \in \mathbb{Q}[x,y,z]$.
Compute an irredundant primary decomposition and the associated primes of the product $\displaystyle P_1P_2P_3$.

How can I compute a primary decomposition???
If you serve solution of this problem, then it's very helpful for me.
an irredundant primary decomposition is: $\displaystyle <x,y> \cap <x,z> \cap <y,z> \cap <x^2,y^2,z^2,xyz>$ and the associated primes are obviously $\displaystyle <x,y>, <x,z>, <y,z>, <x,y,z>.$ the idea is
to first write your ideal as an ideal generated by some monomials. in here we have: $\displaystyle P_1P_2P_3=<x^2y,x^2z,y^2x,y^2z,z^2x,z^2y,xyz>.$ this is equal to the intersection of all ideals generated by
7 monomials, each chosen from one of the generators of $\displaystyle P_1P_2P_3.$ many of these ideals are subsets of some others and so we just delete them. it is important to know that a monomial ideal in
a polynomial ring $\displaystyle k[x_1,x_2, \cdots , x_n],$ where $\displaystyle k$ is a field, is primary if and only if it's in the form $\displaystyle <x_{j_1}^{n_1}, x_{j_2}^{n_2}, \cdots , x_{j_r}^{n_r}, u_1, u_2, \cdots , u_s>,$ where $\displaystyle u_1, u_2, \cdots , u_s$ are monomials in $\displaystyle x_{j_1} , x_{j_2}, \cdots , x_{j_r}.$ for
example in $\displaystyle k[x,y,z],$ the ideal $\displaystyle <x^2,y^2,z^2,xyz>$ is primary but $\displaystyle <x^2,y^2, yz>$ is not primary.