Decomposition of a monomial ideal

Question

I have to find a primary decomposition of the following ideal and I proceeded in this way:

$$(x^2z,x^2y^3,xt^2)=(x)\cap(t^2,x^2z,x^2y^3)=(x)\cap(t^2,x^2)\cap(t^2,z,z^2y^3)=(x)\cap(t^2,x^2)\cap(t^2,z,y^3)\cap(t^2,z,x^2)$$

Now I have to show that $(t^2,z,x^2)$ is redundant and I can prove that fact saying that it contains $(t^2,x^2)$.

How can I show (in a simple way) that other ideals are irredundant?

Answer 1

You can use evaluation.
Lets prove $x\notin (t^2,x^2):$ otherwise $x=t^2f+x^2g.$ Evaluation of $t$, $x$ at $0$, $x$ leads to a contradiction.
Using appropriate evaluation you can say $z\notin (t^2,x^2), $ and so on.