I'm trying to learn the process of primary decomposition of a monomial ideal and struggling. I can't seem to find a resource where I'm able to follow the steps in an example and see the thinking behind them. If, say, we have the ideal $I=\langle x^2, xy, x^2z^2,yz^2\rangle$, the first step I take is to write down the intersection of 8 ideals (can anyone define what the intersection of two ideals actually is?), along the lines of $$I=\langle x^2, x, x^2,y\rangle \cap \langle x^2, y, x^2, y\rangle \cap \dots$$
I'm not sure what the next step is... I start taking the "repeats" out but I'm not sure what the criteria is or what I'm looking for. Can anyone take me through the steps in actually finding a primary decomposition, or point me in the direction of a paper which brings it back down to a basic level? Really lost right now.
The book "Monomial Ideals" by Herzog-Hibi, Section 1-3:
Theorem 1.3.1 and Example 1.3.3 are (probably) what you want. I add image of Theorem and part of proof:
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Note that by Corollary 1.3.2, A monomial ideal is irreducible if and only if it is generated by pure powers of the variables.
As authors say: The proof of Theorem 1.3.1 shows us how we can find such a presentation. The following example illustrates the procedure.
$$ I=(x^2, xy, x^2z^2,yz^2)= \\ (x^2, xy, yz^2)=\\ (x^2, x, yz^2)\cap (x^2, y, yz^2)=\\ (x,yz^2)\cap (x^2, y)=\\ (x,y)\cap (x,z^2) \cap (x^2, y)=\\ (x,z^2) \cap (x^2, y) $$