Primary and irreducible ideals

Question

I have to verify that the ideal $I = \langle x^3,x^2y,xy^3,y^5\rangle \subset R=\mathbb{C}[x,y,z]$ is primary. I then have to go on to show that it is not irreducible by writing it as an intersection of two larger ideals.

Now I have a definition of primary but I am not sure how to actually show that for $f,g \in R$ that if $fg \in I$ then $f \in I$ or $g^m \in I$ for some $m>0$. I am also confused as to how I should reduce the ideal, as I'm not familiar with the process of finding $I_1, I_2$ such that $ I = I_1 \cap I_2$ although I understand it has something to do with splitting products, but is it true that, for instance, $I = <x^3,x^2,xy^3,y^5> \cap <x^3,y,xy^3,y^5> $ for instance?

Answer 1

It is not irreducible.
page.9 of Herzog-Hibi's book, "Monomial Ideals", has:

Corollary $1.3.2$. A monomial ideal is irreducible if and only if it is generated by pure powers of the variables.

After proof, by an example, (1.3.3), they illustrate the procedure. It is very helpful.

Answer 2

$\langle x^3,\underline{x^2y},xy^3,y^5\rangle=\langle x^3,{\color{red}{x^2}},xy^3,y^5\rangle\cap \langle x^3,{\color{red}y},xy^3,y^5\rangle=\langle x^2,\underline{xy^3},y^5\rangle\cap \langle x^3,y\rangle\ (\text{for reducibility you can stop here!})=\langle x^2,{\color{red}x},y^5\rangle\cap\langle x^2,{\color{red}{y^3}},y^5\rangle\cap \langle x^3,y\rangle=\langle x,y^5\rangle\cap\langle x^2,y^3\rangle\cap \langle x^3,y\rangle$

It follows that $I$ is an intersection of three $(x,y)$-primary ideals, so $I$ is primary and is not irreducible.

Remark. If you want to prove directly that your ideal is primary, then notice that $\sqrt I=(x,y)$ which is a maximal ideal in $\mathbb C[x,y]$.