Consider the ideal $J=(xy,yz,zx)$ in $R=\mathbb C[x,y,z]$. As seen here Associated primes of the square of a monomial ideal , $Ass_R (R/J^n)=\{(x,y); (y,z);(z,x);(x,y,z)\}, \forall n \ge 2$.
My question is : What is an irredundant primary decomposition of $J^n$, for $n\ge 2$ ?
Definitely $J^n$ will be the intersection of four primary ideals and the primary components corresponding to $(x,y); (y,z)$ and $(z,x)$ will be unique because these are the minimal primes. Apart from this, I am not sure how to actually get a decomposition.
Please help.
For any subset $S \subseteq V := \{x,y,z\}$ write $I_S$ for the ideal generated by the variables in $S$. Then a few computations with Macaulay2 will start to convince you that the primary decomposition of $J^n$ consists of four ideals: $(I_S)^n$ for each subset $S$ of size two and the ideal $K + K'$ where $K = \langle x^n, y^n, z^n\rangle$ and $K'$ is generated by all monomials $x^ay^bz^c$ of degree $2n$ such that $0 < a,b,c < n$.
To prove this is the irredundant primary decomposition first prove that $J^n$ is the ideal generated by all monomials $x^ay^bz^c$ of degree $2n$ such that $0 \le a,b,c \le n$. Then apply the algorithm in the proof of Lemma 5.18 in Combinatorial Commutative Algebra.