1. ## Primary decomposition

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

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

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: $ \cap \cap \cap $ and the associated primes are obviously $, , , .$ the idea is
to first write your ideal as an ideal generated by some monomials. in here we have: $P_1P_2P_3=.$ this is equal to the intersection of all ideals generated by
7 monomials, each chosen from one of the generators of $P_1P_2P_3.$ 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 $k[x_1,x_2, \cdots , x_n],$ where $k$ is a field, is primary if and only if it's in the form $,$ where $u_1, u_2, \cdots , u_s$ are monomials in $x_{j_1} , x_{j_2}, \cdots , x_{j_r}.$ for
example in $k[x,y,z],$ the ideal $$ is primary but $$ is not primary.